Content-Type: multipart/mixed; boundary="-------------9903040817165" This is a multi-part message in MIME format. ---------------9903040817165 Content-Type: text/plain; name="99-71.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-71.keywords" Jacobi polynomials, hypergeometric functions ---------------9903040817165 Content-Type: application/postscript; name="new1.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="new1.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %%Title: new.dvi %%Pages: 17 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips new.dvi -o new.ps %DVIPSParameters: dpi=300, comments removed %DVIPSSource: TeX output 1999.03.04:1704 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 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2696 71 2 v 790 2742 a(@)s(\022)842 2749 y Fi(2)874 2707 y Fo(sin)934 2687 y Fl(n)p Fk(\000)p Fi(3)1011 2707 y Fn(\022)1034 2714 y Fi(2)1067 2674 y Fn(@)s(v)p 1058 2696 V 1058 2742 a(@)s(\022)1110 2749 y Fi(2)1145 2707 y Fo(+)11 b Fn(l)q Fo(\()p Fn(l)h Fo(+)f Fn(n)g Fg(\000)g Fo(2\))p Fn(v)k Fo(=)f(0)p Fn(:)p eop %%Page: 6 6 6 5 bop 73 203 a Fo(F)l(or)17 b(solutions)f(dep)q(ending)h(on)f(only)g Fn(\022)796 210 y Fi(1)816 203 y Fn(;)g Fo(equation)g(\(2.15\))h (reduces)e(to)345 304 y(1)p 267 327 180 2 v 267 374 a(sin)327 354 y Fl(n)p Fk(\000)p Fi(2)404 374 y Fn(\022)427 381 y Fi(1)478 304 y Fn(d)p 457 327 68 2 v 457 372 a(d\022)505 379 y Fi(1)538 338 y Fo(sin)598 318 y Fl(n)p Fk(\000)p Fi(2)674 338 y Fn(\022)697 345 y Fi(1)722 304 y Fn(dv)771 311 y Fi(0)p 722 327 69 2 v 722 372 a Fn(d\022)770 379 y Fi(1)807 338 y Fo(+)c Fn(l)q Fo(\()p Fn(l)g Fo(+)g Fn(n)g Fg(\000)g Fo(2\))p Fn(v)1123 345 y Fi(0)1156 338 y Fo(=)j(0)p Fn(;)57 b Fo(0)14 b Fn(<)g(\022)1416 345 y Fi(1)1449 338 y Fn(<)g(\031)r(:)220 b Fo(\(2.16\))73 473 y(The)15 b(solution)f(of)h(\(2.16\))g(whic)o(h)f(is)g(b)q(ounded)h (at)g(b)q(oth)g Fn(\022)1123 480 y Fi(1)1156 473 y Fo(=)f(0)h(and)g Fn(\022)1363 480 y Fi(1)1396 473 y Fo(=)f Fn(\031)i Fo(exists)e(if)g Fn(l)h Fo(is)f(in)o(teger)0 533 y(and)j(ma)o(y)e(b)q(e)h(expressed)g (in)f(terms)g(of)i(Gegen)o(bauer)f(p)q(olynomial)711 653 y Fn(v)735 660 y Fi(0)p Fl(;l)789 653 y Fo(=)e Fn(C)880 628 y Fi(\()p Fl(n)p Fk(\000)p Fi(2\))p Fl(=)p Fi(2)876 667 y Fl(l)1011 653 y Fo(\(cos)8 b Fn(\022)1126 660 y Fi(1)1146 653 y Fo(\))p Fn(:)73 740 y Fo(By)13 b(separating)h(the)g(v)m (ariable)f Fn(\022)665 747 y Fi(2)698 740 y Fo(w)o(e)g(obtain)h (equation)g(\(2.16\))g(in)f(whic)o(h)g Fn(n)h Fo(is)f(replaced)g(b)o(y) g Fn(n)6 b Fg(\000)g Fo(1)0 801 y(with)16 b(the)g(eigenfunction)519 921 y Fn(y)543 928 y Fl(q)562 921 y Fo(\()p Fn(\022)604 928 y Fi(2)623 921 y Fo(\))e(=)g Fn(C)747 900 y Fi(\()p Fl(n)p Fk(\000)p Fi(3\))p Fl(=)p Fi(2)743 933 y Fl(q)878 921 y Fo(\(cos)8 b 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Fo(\))1119 2560 y Fk(\000)p Fl(\025)1170 2581 y Fn(;)56 b Fg(j)p Fn(z)r Fg(j)14 b Fn(<)f Fo(1)p Fn(:)73 2736 y Fo(T)l(ransfer)22 b(of)f(the)g(singularit)o(y)g(b)o(y)g(the)g(use)g (of)g(rotation)h(\(2.14\))g(giv)o(es)f(the)g(addition)g(form)o(ula)0 2796 y(\(2.1\).)p eop %%Page: 7 7 7 6 bop 0 203 a Fp(3.)22 b(Bispherical)j(co)r(ordinates.)32 b(Adjoin)n(t)24 b(Jacobi)f(functions.)33 b(Asymmet-)75 278 y(ric)23 b(addition)h(form)n(ula.)0 388 y Fo(Let)16 b(us)h(in)o(tro)q(duce)f(bispherical)f(co)q(ordinates)i(b)o(y)f(the)g (use)g(of)h(the)f(c)o(hain)f(of)i(relations)493 498 y Fn(r)515 505 y Fi(1)548 498 y Fo(=)d Fn(r)c Fo(cos)e Fn(\013;)57 b(r)829 505 y Fi(2)863 498 y Fo(=)14 b Fn(r)9 b Fo(sin)g Fn(\013;)57 b Fo(0)14 b Fg(\024)f Fn(\013)i Fg(\024)e Fn(\031)r(=)p Fo(2)p Fn(:)393 b Fo(\(3.1\))73 608 y Fn(\032)98 615 y Fi(1)p Fl(;)p Fi(0)159 608 y Fo(=)14 b Fn(r)233 615 y Fi(1)253 608 y Fo(;)56 b Fn(x)351 615 y Fi(1)p Fl(;k)414 608 y Fo(=)14 b Fn(\032)491 615 y Fi(1)p Fl(;k)q Fk(\000)p Fi(1)593 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y Fn(\013)1373 1905 y(@)s(u)p 1371 1928 V 1371 1973 a(@)s(\013)1448 1939 y Fo(+)470 2060 y(+)589 2026 y(1)p 513 2048 177 2 v 513 2094 a Fn(r)536 2080 y Fi(2)564 2094 y Fo(cos)630 2080 y Fi(2)658 2094 y Fn(\013)694 2060 y Fo([)809 2026 y(1)p 713 2048 217 2 v 713 2096 a(sin)772 2076 y Fl(m)p Fk(\000)p Fi(2)859 2096 y Fn(\022)882 2103 y Fi(1)p Fl(;)p Fi(1)974 2026 y Fn(@)p 939 2048 99 2 v 939 2094 a(@)s(\022)991 2101 y Fi(1)p Fl(;)p Fi(1)1051 2060 y Fo(sin)1110 2039 y Fl(m)p Fk(\000)p Fi(2)1197 2060 y Fn(\022)1220 2067 y Fi(1)p Fl(;)p Fi(1)1293 2026 y Fn(@)s(u)p 1272 2048 V 1272 2094 a(@)s(\022)1324 2101 y Fi(1)p Fl(;)p Fi(1)1386 2060 y Fo(+)470 2194 y(+)692 2161 y(1)p 513 2183 383 2 v 513 2230 a(sin)573 2210 y Fi(2)601 2230 y Fn(\022)624 2237 y Fi(1)p Fl(;)p Fi(1)679 2230 y Fo(sin)739 2210 y Fl(m)p Fk(\000)p Fi(3)826 2230 y Fn(\022)849 2237 y Fi(1)p Fl(;)p Fi(2)940 2161 y Fn(@)p 906 2183 99 2 v 906 2229 a(@)s(\022)958 2236 y Fi(1)p Fl(;)p Fi(2)1017 2194 y Fo(sin)1077 2174 y Fl(m)p Fk(\000)p Fi(3)1164 2194 y Fn(\022)1187 2201 y Fi(1)p Fl(;)p Fi(2)1260 2161 y Fn(@)s(u)p 1239 2183 V 1239 2229 a(@)s(\022)1291 2236 y Fi(1)p Fl(;)p Fi(2)1353 2194 y Fo(+)i Fn(:::)f Fo(+)298 b(\(3.5\))470 2334 y(+)795 2300 y(1)p 513 2323 590 2 v 513 2370 a(sin)573 2350 y Fi(2)601 2370 y Fn(\022)624 2377 y Fi(1)p Fl(;)p Fi(1)679 2370 y Fo(sin)739 2350 y Fi(2)767 2370 y Fn(\022)790 2377 y Fi(1)p Fl(;)p Fi(2)837 2370 y Fn(:::)8 b Fo(sin)946 2350 y Fi(2)974 2370 y Fn(\022)997 2377 y Fi(1)p Fl(;m)p Fk(\000)p Fi(2)1139 2300 y Fn(@)1168 2282 y Fi(2)1187 2300 y Fn(u)p 1112 2323 130 2 v 1112 2368 a(@)s(\022)1165 2351 y Fi(2)1164 2380 y Fl(m)p Fk(\000)p Fi(1)1247 2334 y Fo(])i(+)470 2469 y(+)586 2435 y(1)p 513 2457 171 2 v 513 2505 a Fn(r)536 2490 y Fi(2)564 2505 y Fo(sin)624 2485 y Fi(2)652 2505 y Fn(\013)689 2469 y Fo([)798 2435 y(1)p 708 2457 207 2 v 708 2505 a(sin)767 2485 y Fl(n)p Fk(\000)p Fi(2)844 2505 y Fn(\022)867 2512 y Fi(2)p Fl(;)p Fi(1)959 2435 y Fn(@)p 924 2457 99 2 v 924 2503 a(@)s(\022)976 2510 y Fi(2)p Fl(;)p Fi(1)1035 2469 y Fo(sin)1095 2448 y Fl(n)p Fk(\000)p Fi(2)1172 2469 y Fn(\022)1195 2476 y Fi(2)p Fl(;)p Fi(1)1268 2435 y Fn(@)s(u)p 1247 2457 V 1247 2503 a(@)s(\022)1299 2510 y Fi(2)p Fl(;)p Fi(1)1361 2469 y Fo(+)470 2603 y(+)687 2570 y(1)p 513 2592 373 2 v 513 2639 a(sin)573 2619 y Fi(2)601 2639 y Fn(\022)624 2646 y Fi(2)p Fl(;)p Fi(1)679 2639 y Fo(sin)739 2619 y Fl(n)p Fk(\000)p Fi(3)816 2639 y Fn(\022)839 2646 y Fi(2)p Fl(;)p Fi(2)931 2570 y Fn(@)p 896 2592 99 2 v 896 2638 a(@)s(\022)948 2645 y Fi(2)p Fl(;)p Fi(2)1007 2603 y Fo(sin)1067 2583 y Fl(n)p Fk(\000)p Fi(3)1144 2603 y Fn(\022)1167 2610 y Fi(2)p Fl(;)p Fi(2)1240 2570 y Fn(@)s(u)p 1219 2592 V 1219 2638 a(@)s(\022)1271 2645 y Fi(2)p Fl(;)p Fi(2)1333 2603 y Fo(+)h Fn(:::)f Fo(+)470 2743 y(+)791 2709 y(1)p 513 2731 580 2 v 513 2779 a(sin)573 2759 y Fi(2)601 2779 y Fn(\022)624 2786 y Fi(2)p Fl(;)p Fi(1)679 2779 y Fo(sin)739 2759 y Fi(2)767 2779 y Fn(\022)790 2786 y Fi(2)p Fl(;)p Fi(2)837 2779 y Fn(:::)e Fo(sin)946 2759 y Fi(2)974 2779 y Fn(\022)997 2786 y Fi(2)p Fl(;n)p Fk(\000)p Fi(2)1124 2709 y Fn(@)1153 2691 y Fi(2)1173 2709 y Fn(u)p 1102 2731 120 2 v 1102 2777 a(@)s(\022)1155 2760 y Fi(2)1154 2789 y Fl(n)p Fk(\000)p Fi(1)1227 2743 y Fo(])14 b(=)f(0)p eop %%Page: 8 8 8 7 bop 73 203 a Fo(Equation)17 b(\(3.5\))f(a)o(v)o(eraged)g(o)o(v)o (er)g(the)g(angles)g Fn(\022)960 210 y Fi(1)p Fl(;)p Fi(2)1007 203 y Fn(;)8 b(:::;)g(\022)1116 210 y Fi(1)p Fl(;m)p Fk(\000)p Fi(1)1220 203 y Fn(;)57 b(\022)1314 210 y Fi(2)p Fl(;)p Fi(2)1360 203 y Fn(;)8 b(:::;)g(\022)1469 210 y Fi(2)p Fl(;n)p Fk(\000)p Fi(1)1580 203 y Fo(reduces)15 b(to)473 334 y Fn(Lu)f Fg(\021)669 300 y Fo(1)p 605 322 151 2 v 605 368 a Fn(r)628 354 y Fl(m)p Fi(+)p Fl(n)p Fk(\000)p Fi(1)778 300 y Fn(@)p 766 322 52 2 v 766 368 a(@)s(r)823 334 y(r)846 313 y Fl(m)p Fi(+)p Fl(n)p Fk(\000)p Fi(1)978 300 y Fn(@)s(u)p 978 322 57 2 v 980 368 a(@)s(r)1051 334 y Fo(+)473 453 y(+)709 419 y(1)p 516 442 412 2 v 516 489 a Fn(r)539 474 y Fi(2)567 489 y Fo(cos)633 474 y Fl(m)p Fk(\000)p Fi(1)719 489 y Fn(\013)9 b Fo(sin)819 469 y Fl(n)p Fk(\000)p Fi(1)896 489 y Fn(\013)953 419 y(@)p 937 442 60 2 v 937 487 a(@)s(\013)1010 453 y Fo(cos)1075 432 y Fl(m)p Fk(\000)p Fi(1)1162 453 y Fn(\013)g Fo(sin)1262 432 y Fl(n)p Fk(\000)p Fi(1)1339 453 y Fn(\013)1377 419 y(@)s(u)p 1375 442 V 1375 487 a(@)s(\013)1451 453 y Fo(+)301 b(\(3.6\))473 574 y(+)592 540 y(1)p 516 562 177 2 v 516 608 a Fn(r)539 594 y Fi(2)567 608 y Fo(cos)633 594 y Fi(2)661 608 y Fn(\013)775 540 y Fo(1)p 702 562 172 2 v 702 610 a(sin)762 590 y Fl(m)p Fk(\000)p Fi(2)849 610 y Fn(\022)896 540 y(@)p 883 562 54 2 v 883 608 a(@)s(\022)950 574 y Fo(sin)1010 553 y Fl(m)p Fk(\000)p Fi(2)1096 574 y Fn(\022)1125 540 y(@)s(u)p 1125 562 57 2 v 1126 608 a(@)s(\022)1198 574 y Fo(+)473 702 y(+)589 668 y(1)p 516 691 171 2 v 516 738 a Fn(r)539 723 y Fi(2)567 738 y Fo(sin)627 718 y Fi(2)655 738 y Fn(\013)770 668 y Fo(1)p 697 691 V 697 738 a(sin)756 718 y Fl(n)p Fk(\000)p Fi(2)833 738 y Fn( )894 668 y(@)p 877 691 62 2 v 877 736 a(@)s( )952 702 y Fo(sin)1012 681 y Fl(n)p Fk(\000)p Fi(2)1089 702 y Fn( )1130 668 y(@)s(u)p 1128 691 V 1128 736 a(@)s( )1208 702 y Fo(=)14 b(0)p Fn(;)73 830 y Fo(where)i(w)o(e)g(put)g Fn(\022)f Fo(=)f Fn(\022)488 837 y Fi(1)p Fl(;)p Fi(1)535 830 y Fn(;)57 b( )15 b Fo(=)f Fn(\022)728 837 y Fi(2)p Fl(;)p Fi(1)775 830 y Fo(.)73 891 y(In)i(Cartesian)h(co)q(ordinates)168 995 y Fn(x)196 1002 y Fi(1)229 995 y Fo(=)d(cos)9 b Fn(\013)f Fo(cos)h Fn(\022)q(;)57 b(x)591 1002 y Fi(2)624 995 y Fo(=)14 b(cos)8 b Fn(\013)h Fo(sin)f Fn(\022)q(;)57 b(x)980 1002 y Fi(3)1013 995 y Fo(=)14 b(sin)8 b Fn(\013)h Fo(cos)f Fn( )r(;)105 b(x)1427 1002 y Fi(4)1460 995 y Fo(=)14 b(sin)8 b Fn(\013)h Fo(sin)f Fn( )0 1099 y Fo(Equation)17 b(\(3.6\))f(rearranges)h(to)g(the)f(form)384 1249 y Fn(Lu)e Fo(=)538 1187 y Fi(4)511 1201 y Fj(X)518 1306 y Fl(i)p Fi(=1)596 1215 y Fn(@)625 1197 y Fi(2)644 1215 y Fn(u)p 596 1237 76 2 v 596 1283 a(@)s(x)653 1266 y Fi(2)653 1296 y Fl(i)688 1249 y Fo(+)d(\()p Fn(m)g Fg(\000)g Fo(2\))919 1215 y(1)p 908 1237 48 2 v 908 1283 a Fn(x)936 1290 y Fi(2)975 1215 y Fn(@)s(u)p 965 1237 76 2 v 965 1283 a(@)s(x)1022 1290 y Fi(2)1057 1249 y Fo(+)g(\()p Fn(n)g Fg(\000)g Fo(2\))1275 1215 y(1)p 1263 1237 48 2 v 1263 1283 a Fn(x)1291 1290 y Fi(4)1331 1215 y Fn(@)s(u)p 1321 1237 76 2 v 1321 1283 a(@)s(x)1378 1290 y Fi(4)1416 1249 y Fo(=)i(0)p Fn(:)285 b Fo(\(3.7\))73 1401 y(Equation)13 b(\(3.6\))g(allo)o(ws)g (separation)h(of)f(v)m(ariables.)20 b(Substitution)12 b Fn(r)1338 1383 y Fl(l)1352 1401 y Fn(v)r Fo(\()p Fn(\013;)c(\022)q(;) g( )r Fo(\))j(yields)h(the)h(equa-)0 1461 y(tion)j(for)h (\(bi\)spherical)e(functions)h(whic)o(h)g(has)h(the)f(form)548 1556 y(1)p 380 1579 360 2 v 380 1626 a(cos)445 1612 y Fl(m)p Fk(\000)p Fi(1)532 1626 y Fn(\013)9 b Fo(sin)631 1606 y Fl(n)p Fk(\000)p Fi(1)708 1626 y Fn(\013)765 1556 y(@)p 750 1579 60 2 v 750 1624 a(@)s(\013)823 1590 y Fo(cos)888 1570 y Fl(m)p Fk(\000)p Fi(1)975 1590 y Fn(\013)f Fo(sin)1074 1570 y Fl(n)p Fk(\000)p Fi(1)1151 1590 y Fn(\013)1190 1556 y(@)s(v)p 1187 1579 V 1187 1624 a(@)s(\013)1263 1590 y Fo(+)375 1711 y(+)468 1677 y(1)p 418 1699 125 2 v 418 1745 a(cos)483 1731 y Fi(2)511 1745 y Fn(\013)626 1677 y Fo(1)p 553 1699 172 2 v 553 1747 a(sin)612 1727 y Fl(m)p Fk(\000)p Fi(2)699 1747 y Fn(\022)746 1677 y(@)p 734 1699 54 2 v 734 1745 a(@)s(\022)800 1711 y Fo(sin)860 1690 y Fl(m)p Fk(\000)p Fi(2)947 1711 y Fn(\022)976 1677 y(@)s(v)p 976 1699 V 976 1745 a(@)s(\022)1046 1711 y Fo(+)706 b(\(3.8\))375 1839 y(+)465 1805 y(1)p 418 1828 120 2 v 418 1875 a(sin)478 1855 y Fi(2)506 1875 y Fn(\013)620 1805 y Fo(1)p 547 1828 171 2 v 547 1875 a(sin)607 1855 y Fl(n)p Fk(\000)p Fi(2)684 1875 y Fn( )744 1805 y(@)p 727 1828 62 2 v 727 1873 a(@)s( )803 1839 y Fo(sin)862 1819 y Fl(n)p Fk(\000)p Fi(2)939 1839 y Fn( )982 1805 y(@)s(v)p 978 1828 V 978 1873 a(@)s( )1056 1839 y Fo(+)11 b Fn(p)p Fo(\()p Fn(p)h Fo(+)f Fn(m)g Fo(+)g Fn(n)g Fg(\000)g Fo(2\))p Fn(v)k Fo(=)f(0)p Fn(:)73 1970 y Fo(F)l(or)j(the)f(solution)g (dep)q(ending)g(on)h Fn(\013)g Fo(alone,)f(\(3.8\))g(reduces)g(to)425 2066 y(1)p 257 2088 360 2 v 257 2135 a(cos)322 2121 y Fl(m)p Fk(\000)p Fi(1)409 2135 y Fn(\013)9 b Fo(sin)509 2115 y Fl(n)p Fk(\000)p Fi(1)586 2135 y Fn(\013)642 2066 y(@)p 627 2088 60 2 v 627 2134 a(@)s(\013)700 2099 y Fo(cos)765 2079 y Fl(m)p Fk(\000)p Fi(1)852 2099 y Fn(\013)g Fo(sin)951 2079 y Fl(n)p Fk(\000)p Fi(1)1028 2099 y Fn(\013)1064 2066 y(@)s(v)1117 2073 y Fi(0)p 1064 2088 72 2 v 1071 2134 a Fn(@)s(\013)1152 2099 y Fo(+)i Fn(p)p Fo(\()p Fn(p)h Fo(+)f Fn(m)g Fo(+)g Fn(n)g Fg(\000)g Fo(2\))p Fn(v)1589 2106 y Fi(0)1623 2099 y Fo(=)i(0)p Fn(;)78 b Fo(\(3.9\))252 2189 y(0)14 b Fn(<)g(\013)g(<)g(\031)r(=)p Fo(2)p Fn(:)73 2294 y Fo(The)19 b(solution)g(of)f(\(3.9\))h(b)q(ounded) h(at)f Fn(\013)f Fo(=)f(0)p Fn(;)8 b(\031)r(=)p Fo(2)20 b(exists)d(if)i Fn(p)f Fo(=)f(2)p Fn(q)r(;)57 b(q)20 b Fo(is)e(in)o(teger,)g(and)h(has)0 2354 y(the)d(form)377 2474 y Fn(v)401 2481 y Fl(o)433 2474 y Fo(=)e Fn(P)523 2454 y Fm({)q Fl(;\025)516 2486 y(q)581 2474 y Fo(\(cos)8 b(2)p Fn(\013)p Fo(\))p Fn(;)57 b Fh({)16 b Fo(=)e(\()p Fn(n)d Fg(\000)g Fo(2\))p Fn(=)p Fo(2)p Fn(;)58 b(\025)14 b Fo(=)g(\()p Fn(m)d Fg(\000)g Fo(2\))p Fn(=)p Fo(2)p Fn(:)73 2559 y Fo(By)17 b(separating)h(the)f(v)m(ariables)h Fn(\022)g Fo(and)g Fn( )h Fo(w)o(e)e(come)f(to)i(Sturm-Liouville)d (problems)h(similar)f(to)0 2619 y(\(2.18\))i(with)f(the)g(follo)o(wing) g(solutions)451 2730 y(\002)489 2737 y Fl(l)502 2730 y Fo(\()p Fn(\022)q Fo(\))e(=)g Fn(C)669 2705 y Fi(\()p Fl(m)p Fk(\000)p Fi(2\))p Fl(=)p Fi(2)665 2744 y Fl(l)809 2730 y Fo(\(cos)9 b Fn(\022)q Fo(\))p Fn(;)451 2811 y Fo(\011)489 2818 y Fl(k)510 2811 y Fo(\()p Fn( )r Fo(\))14 b(=)f Fn(C)686 2786 y Fi(\()p Fl(n)p Fk(\000)p Fi(2\))p Fl(=)p Fi(2)682 2825 y Fl(k)817 2811 y Fo(\(cos)c Fn( )r Fo(\))p Fn(;)56 b(l)14 b Fo(=)g(0)p Fn(;)8 b Fo(1)p Fn(;)g(:::;)56 b(k)16 b Fo(=)d(0)p Fn(;)8 b Fo(1)p Fn(:::)p eop %%Page: 9 9 9 8 bop 73 203 a Fo(The)16 b(bispherical)g(function,)f(i.e.)20 b(the)c(b)q(ounded)h(solution)g(of)f(\(3.8\))h(is)f(expressed)g(in)g (the)g(form)638 290 y Fn(v)r Fo(\()p Fn(\013;)8 b(\022)q(;)g( )r Fo(\))13 b(=)h Fn(y)r Fo(\()p Fn(\013)p Fo(\)\002)1033 297 y Fl(l)1045 290 y Fo(\()p Fn(\022)q Fo(\)\011)1145 297 y Fl(k)1167 290 y Fo(\()p Fn( )r Fo(\))p Fn(:)512 b Fo(\(3.10\))73 376 y(Substituting)17 b(\(3.10\))f(in)g(\(3.8\))h(w)o (e)f(obtain)g(\(cf.)21 b([11,)16 b(p.)g(475]\))h(the)f(equation)543 453 y(1)p 375 476 360 2 v 375 523 a(cos)441 508 y Fl(m)p Fk(\000)p Fi(1)527 523 y Fn(\013)9 b Fo(sin)627 503 y Fl(n)p Fk(\000)p Fi(1)704 523 y Fn(\013)761 453 y(d)p 745 476 57 2 v 745 521 a(d\013)815 487 y Fo(cos)880 466 y Fl(m)p Fk(\000)p Fi(1)967 487 y Fn(\013)g Fo(sin)1067 466 y Fl(n)p Fk(\000)p Fi(1)1143 487 y Fn(\013)1182 453 y(dy)p 1179 476 V 1179 521 a(d\013)1253 487 y Fo(+)474 b(\(3.11\))370 611 y(+[)p Fn(p)p Fo(\()p Fn(p)12 b Fo(+)f Fn(m)g Fo(+)g Fn(n)g Fg(\000)g Fo(2\))g Fg(\000)852 577 y Fn(l)q Fo(\()p Fn(l)g Fo(+)g Fn(m)g Fg(\000)g Fo(2\))p 852 599 258 2 v 918 645 a(cos)983 630 y Fi(2)1011 645 y Fn(\013)1125 611 y Fg(\000)1180 577 y Fn(k)r Fo(\()p Fn(k)i Fo(+)e Fn(n)g Fg(\000)g Fo(2\))p 1180 599 267 2 v 1254 646 a(sin)1313 626 y Fi(2)1341 646 y Fn(\013)1451 611 y Fo(])p Fn(y)k Fo(=)f(0)p Fn(;)370 700 y Fo(0)h Fn(<)e(\013)i(<)e(\031)r(=)p Fo(2)p Fn(;)73 787 y Fo(whic)o(h)j(is)h (natural)g(to)g(call)f(as)i(the)e(equation)h(for)g(Jacobi)g(adjoin)o(t) g(functions.)23 b(These)17 b(functions)0 847 y(ha)o(v)o(e)e(the)h(form) 539 907 y Fn(y)565 886 y Fl(k)q(;l)563 919 y(q)607 907 y Fo(\()p Fn(\013)p Fo(\))e(=)g(sin)802 886 y Fl(k)831 907 y Fn(\013)9 b Fo(cos)936 886 y Fl(l)958 907 y Fn(\013P)1027 883 y Fm({)q Fi(+)p Fl(k)q(;\025)p Fi(+)p Fl(l)1020 918 y(q)q Fk(\000)p Fl(s)1170 907 y Fo(\(cos)f(2)p Fn(\013)p Fo(\))p Fn(;)415 b Fo(\(3.12\))73 984 y(where)16 b Fn(k)d Fo(+)e Fn(l)k Fo(=)e(2)p Fn(s;)8 b(s)14 b Fo(=)g(0)p Fn(;)8 b Fo(1)p Fn(;)g(:::;)g(q)17 b Fo(if)f Fn(p)e Fo(=)g(2)p Fn(q)k Fo(and)f Fn(k)c Fo(+)e Fn(l)j Fo(=)g(2)p Fn(s)d Fo(+)g(1)17 b(if)f Fn(p)e Fo(=)g(2)p Fn(q)f Fo(+)e(1.)73 1044 y(F)l(rom)16 b(the)g(form)o(ulae)f(written)h(ab)q(o)o(v)o(e)h(it)f (follo)o(ws)h(that)g(biaxially-symm)o(etri)o(c)d(solution)i(of)i (\(3.6\))0 1104 y(with)e(the)g(singularit)o(y)g(at)g Fn(\013)e Fo(=)g(0)p Fn(;)57 b(r)15 b Fo(=)f Fn(r)799 1117 y Fi(0)835 1104 y Fo(can)i(b)q(e)h(written)f(in)f(the)h(form)593 1166 y Fk(1)575 1181 y Fj(X)580 1286 y Fl(q)q Fi(=0)655 1229 y Fn(A)692 1236 y Fl(q)711 1229 y Fo(\()744 1195 y Fn(r)p 735 1217 42 2 v 735 1263 a(r)757 1270 y Fi(0)781 1229 y Fo(\))800 1208 y Fi(2)p Fl(q)837 1229 y Fn(P)875 1208 y Fm({)q Fl(;\025)868 1241 y(q)933 1229 y Fo(\(cos)8 b(2)p Fn(\013)p Fo(\))p Fn(;)58 b(r)15 b(<)f(r)1282 1236 y Fi(0)1301 1229 y Fn(:)450 b Fo(\(3.13\))73 1365 y(F)l(rom)15 b(the)h(Cartesian)h(form)e(\(3.8\))i(it)e(is)h(seen)g(that)h (Eq.\(3.6\))f(is)g(in)o(v)m(arian)o(t)g(under)g(rotation)194 1485 y Fn(x)222 1465 y Fk(0)222 1498 y Fi(1)255 1485 y Fo(=)e Fn(x)335 1492 y Fi(1)363 1485 y Fo(cos)9 b Fn(\034)16 b Fg(\000)11 b Fn(x)552 1492 y Fi(3)580 1485 y Fo(sin)d Fn(\034)s(;)57 b(x)771 1465 y Fk(0)771 1498 y Fi(3)804 1485 y Fo(=)14 b Fn(x)884 1492 y Fi(1)912 1485 y Fo(sin)8 b Fn(\034)17 b Fo(+)11 b Fn(x)1095 1492 y Fi(3)1122 1485 y Fo(cos)e Fn(\034)s(;)57 b(x)1319 1465 y Fk(0)1319 1498 y Fi(2)1352 1485 y Fo(=)14 b Fn(x)1432 1492 y Fi(2)1451 1485 y Fn(;)57 b(x)1550 1465 y Fk(0)1550 1498 y Fi(4)1583 1485 y Fo(=)14 b Fn(x)1663 1492 y Fi(4)1682 1485 y Fn(:)73 1562 y Fo(This)j(fact)f(is)g(equiv)m(alen)o(t)e(to)j(in)o(v)m(ariance)e (of)i(\(3.6\))f(under)h(transformation)199 1649 y(cos)264 1628 y Fi(2)292 1649 y Fn(\013)323 1628 y Fk(0)377 1649 y Fo(=)41 b(cos)522 1628 y Fi(2)550 1649 y Fn(\013)p Fo(\(cos)665 1628 y Fi(2)693 1649 y Fn(\022)10 b Fo(cos)791 1628 y Fi(2)819 1649 y Fn(\034)17 b Fo(+)870 b(\(3.14\))377 1721 y(+)41 b(sin)516 1701 y Fi(2)544 1721 y Fn(\022)q Fo(\))11 b(+)g(sin)707 1701 y Fi(2)735 1721 y Fn(\013)e Fo(sin)835 1701 y Fi(2)863 1721 y Fn(\034)14 b Fo(cos)963 1701 y Fi(2)991 1721 y Fn( )f Fo(+)e(\(1)p Fn(=)p Fo(2\))d(sin)h(2)p Fn(\013)g Fo(sin)g(2)p Fn(\034)14 b Fo(cos)9 b Fn(\022)g Fo(cos)g Fn( )r(:)463 1858 y Fo(tan)f Fn(\022)565 1838 y Fl(=)627 1858 y Fo(=)958 1825 y(cos)g Fn(\013)h Fo(sin)f Fn(\022)p 711 1847 699 2 v 711 1892 a Fo(cos)h Fn(\013)f Fo(cos)h Fn(\022)g Fo(cos)g Fn(\034)17 b Fg(\000)11 b Fo(sin)d Fn(\013)g Fo(cos)h Fn( )h Fo(sin)e Fn(\034)1415 1858 y(;)336 b Fo(\(3.15\))453 1987 y(tan)9 b Fn( )566 1966 y Fl(=)627 1987 y Fo(=)955 1953 y(sin)f Fn(\013)h Fo(sin)f Fn( )p 711 1976 698 2 v 711 2021 a Fo(cos)h Fn(\013)f Fo(cos)h Fn(\022)g Fo(sin)g Fn(\034)16 b Fo(+)11 b(sin)e Fn(\013)f Fo(cos)h Fn( )h Fo(cos)e Fn(\034)1414 1987 y(:)73 2099 y Fo(By)14 b(transforming)g(the)g(singular)h(solution) f(\(3.13\))h(to)g(v)m(ariables)f(\(3.14\),)h(\(3.15\))h(w)o(e)d(come)g (to)i(the)0 2159 y(addition)h(form)o(ula)147 2271 y Fn(P)185 2250 y Fm({)q Fl(;\025)178 2283 y(n)243 2271 y Fo(\(cos)8 b(2)p Fn(\013)390 2250 y Fk(0)403 2271 y Fo(\))13 b(=)512 2208 y Fl(n)487 2223 y Fj(X)491 2329 y Fl(k)q Fi(=0)593 2208 y Fl(n)567 2223 y Fj(X)575 2329 y Fl(l)p Fi(=0)648 2271 y Fn(b)669 2278 y Fl(n;k)q(;l)742 2271 y Fn(F)781 2250 y Fl(k)q(;l)774 2283 y(n)822 2271 y Fo(\()p Fn(\013)p Fo(\))p Fn(G)929 2250 y Fl(k)q(;l)929 2283 y(n)972 2271 y Fo(\()p Fn(\034)6 b Fo(\))p Fn(C)1076 2245 y Fm({)q Fk(\000)p Fi(1)p Fl(=)p Fi(2)1072 2284 y Fl(l)1182 2271 y Fo(\(cos)j Fn(\022)q Fo(\))p Fn(C)1357 2245 y Fl(\025)p Fk(\000)p Fi(1)p Fl(=)p Fi(2)1353 2284 y Fl(k)1460 2271 y Fo(\(cos)f Fn( )r Fo(\))p Fn(:)146 b Fo(\(3.16\))73 2402 y(F)l(orm)o(ula)11 b(\(3.16\))j(is)f(asymmetri)o(c)d(ab)q(out)k(v) m(ariables)f Fn(\013)g Fo(and)g Fn(\034)19 b Fo(\(b)q(ecause)13 b(of)g(the)g(analogous)h(asym-)0 2462 y(metry)19 b(of)j(transformation) g(\(3.14\)\).)37 b(The)21 b(co)q(e\016cien)o(ts)f Fn(G)1135 2444 y Fl(k)q(;l)1135 2474 y(n)1199 2462 y Fo(are)i(expressed)f(with)g (the)g(use)h(of)f(a)0 2522 y(quadrature)c(along)g(the)f(circle)e(of)j (singularities)509 2638 y Fn(G)547 2618 y Fl(k)q(;l)547 2651 y(n)590 2638 y Fo(\()p Fn(\034)6 b Fo(\))14 b(=)720 2570 y Fj(Z)770 2584 y Fi(2)p Fl(\031)748 2683 y Fi(0)820 2638 y Fn(F)859 2618 y Fl(k)q(;l)852 2651 y(n)900 2638 y Fo(\()p Fn(\013)950 2645 y Fi(1)970 2638 y Fo(\))p Fn(C)1028 2613 y Fm({)q Fk(\000)p Fi(1)p Fl(=)p Fi(2)1024 2652 y Fl(l)1135 2638 y Fo(\(cos)8 b Fn( )1259 2645 y Fi(1)1279 2638 y Fo(\))p Fn(d\022)1347 2618 y Fk(\003)1367 2638 y Fn(;)384 b Fo(\(3.17\))73 2751 y(where)16 b(cos)9 b Fn(\013)319 2758 y Fi(1)352 2751 y Fo(=)14 b(cos)9 b Fn(\022)502 2733 y Fk(\003)530 2751 y Fo(cos)f Fn(\034)s(;)57 b Fo(tan)9 b Fn( )809 2758 y Fi(1)842 2751 y Fo(=)14 b(tan)9 b Fn(\022)997 2733 y Fk(\003)1016 2751 y Fn(=)f Fo(cos)i Fn(\034)s(:)73 2811 y Fo(T)l(o)17 b(\014nd)g(the)f(symme)o (tric)d(addition)j(form)o(ula)f(w)o(e)h(had)h(to)f(consider)g (trispherical)f(co)q(ordinates.)p eop %%Page: 10 10 10 9 bop 0 203 a Fp(4.)22 b(Laplace)c(equation)i(in)f(trispherical)i (co)r(ordinates.)28 b(The)18 b(addition)i(for-)75 278 y(m)n(ula)j(for)f(P)390 256 y Fe({)r Fd(;)p Fc(0)390 293 y Fd(n)454 278 y Fq(\()p Fb(x)p Fq(\))0 388 y Fo(T)l(rispherical)15 b(co)q(ordinates)i(are)g(in)o(tro)q(duced)e(b)o(y)h(the)g(relations)493 498 y Fn(r)515 505 y Fi(1)548 498 y Fo(=)e Fn(r)c Fo(cos)e Fn(\013;)57 b(r)829 505 y Fi(2)863 498 y Fo(=)14 b Fn(r)9 b Fo(sin)g Fn(\013;)57 b Fo(0)14 b Fg(\024)f Fn(\013)i Fg(\024)e Fn(\031)r(=)p Fo(2;)393 b(\(4.1\))455 608 y Fn(r)477 615 y Fi(2)p Fl(;)p Fi(1)538 608 y Fo(=)14 b Fn(r)612 615 y Fi(2)640 608 y Fo(cos)9 b Fn(\014)s(;)56 b(r)837 615 y Fi(2)p Fl(;)p Fi(2)898 608 y Fo(=)13 b Fn(r)971 615 y Fi(2)999 608 y Fo(sin)c Fn(\014)s(;)56 b Fo(0)14 b Fg(\024)f Fn(\014)k Fg(\024)c Fn(\031)r(=)p Fo(2)356 b(\(4.2\))123 695 y Fn(\032)148 702 y Fi(0)182 695 y Fo(=)14 b Fn(r)256 702 y Fi(1)275 695 y Fn(;)57 b(x)374 702 y Fl(k)409 695 y Fo(=)14 b Fn(\032)486 702 y Fl(k)q Fk(\000)p Fi(1)561 695 y Fo(cos)8 b Fn(\022)657 702 y Fl(k)678 695 y Fn(;)57 b(\032)774 702 y Fl(k)809 695 y Fo(=)14 b Fn(\032)886 702 y Fl(k)q Fk(\000)p Fi(1)961 695 y Fo(sin)8 b Fn(\022)1052 702 y Fl(k)1073 695 y Fn(;)57 b(k)16 b Fo(=)d(1)p Fn(;)8 b Fo(2)p Fn(;)g(:::p)j Fg(\000)g Fo(1;)57 b Fn(x)1578 702 y Fl(p)1611 695 y Fo(=)14 b Fn(\032)1688 702 y Fl(p)p Fk(\000)p Fi(1)1753 695 y Fo(;)252 832 y(0)g Fg(\024)f Fn(\022)365 839 y Fl(k)400 832 y Fg(\024)h Fn(\031)r(;)56 b(k)16 b Fo(=)e(1)p Fn(;)8 b Fo(2)p Fn(;)g(:::;)g(p)i Fg(\000)h Fo(2;)57 b(0)14 b Fg(\024)g Fn(\022)1095 839 y Fl(p)p Fk(\000)p Fi(1)1173 832 y Fg(\024)g Fo(2)p Fn(\031)r Fo(;)496 b(\(4.3\))252 904 y Fn(\032)277 911 y Fl(p)310 904 y Fo(=)14 b Fn(r)384 911 y Fi(1)404 904 y Fn(;)56 b(x)502 911 y Fl(p)p Fi(+)p Fl(k)583 904 y Fo(=)13 b Fn(\032)659 911 y Fl(p)p Fi(+)p Fl(k)q Fk(\000)p Fi(1)779 904 y Fo(cos)c Fn(\022)876 911 y Fl(p)p Fi(+)p Fl(k)942 904 y Fn(;)57 b(\032)1038 911 y Fl(p)p Fi(+)p Fl(k)1118 904 y Fo(=)14 b Fn(\032)1195 911 y Fl(p)p Fi(+)p Fl(k)q Fk(\000)p Fi(1)1315 904 y Fo(sin)9 b Fn(\022)1407 911 y Fl(p)p Fi(+)p Fl(k)1473 904 y Fn(;)252 977 y(k)15 b Fo(=)f(1)p Fn(;)8 b Fo(2)p Fn(;)g(:::q)k Fg(\000)f Fo(1;)57 b Fn(x)685 984 y Fl(p)p Fi(+)p Fl(q)762 977 y Fo(=)14 b Fn(\032)839 984 y Fl(p)p Fi(+)p Fl(q)q Fk(\000)p Fi(1)949 977 y Fo(;)252 1050 y(0)g Fg(\024)f Fn(\022)365 1057 y Fl(p)p Fi(+)p Fl(k)445 1050 y Fg(\024)h Fn(\031)r(;)56 b(k)16 b Fo(=)e(1)p Fn(;)8 b Fo(2)p Fn(;)g(:::;)g(q)j Fg(\000)g Fo(2;)57 b(0)14 b Fg(\024)g Fn(\022)1139 1057 y Fl(p)p Fi(+)p Fl(q)q Fk(\000)p Fi(1)1262 1050 y Fg(\024)f Fo(2)p Fn(\031)r Fo(;)408 b(\(4.4\))252 1122 y Fn(\032)277 1129 y Fl(p)p Fi(+)p Fl(q)355 1122 y Fo(=)14 b Fn(r)429 1129 y Fi(2)p Fl(;)p Fi(2)476 1122 y Fn(;)56 b(x)574 1129 y Fl(p)p Fi(+)p Fl(q)q Fi(=)p Fl(k)699 1122 y Fo(=)14 b Fn(\032)776 1129 y Fl(p)p Fi(+)p Fl(q)q Fi(=)p Fl(k)q Fk(\000)p Fi(1)940 1122 y Fo(cos)9 b Fn(\022)1037 1129 y Fl(p)p Fi(+)p Fl(q)q Fi(+)p Fl(k)1147 1122 y Fn(;)252 1195 y(\032)277 1202 y Fl(p)p Fi(+)p Fl(q)q Fi(+)p Fl(k)401 1195 y Fo(=)14 b Fn(\032)478 1202 y Fl(p)p Fi(+)p Fl(q)q Fi(+)p Fl(k)q Fk(\000)p Fi(1)643 1195 y Fo(sin)8 b Fn(\022)734 1202 y Fl(p)p Fi(+)p Fl(q)q Fi(+)p Fl(k)845 1195 y Fn(;)56 b(k)16 b Fo(=)e(1)p Fn(;)8 b Fo(2)p Fn(;)g(:::s)i Fg(\000)h Fo(1;)57 b Fn(x)1348 1202 y Fl(p)p Fi(+)p Fl(q)q Fi(+)p Fl(s)1469 1195 y Fo(=)14 b Fn(\032)1546 1202 y Fl(p)p Fi(+)p Fl(q)q Fi(+)p Fl(s)p Fk(\000)p Fi(1)1699 1195 y Fo(;)252 1268 y(0)g Fg(\024)f Fn(\022)365 1275 y Fl(p)p Fi(+)p Fl(q)q Fi(+)p Fl(k)490 1268 y Fg(\024)g Fn(\031)r(;)57 b(k)15 b Fo(=)f(1)p Fn(;)8 b Fo(2)p Fn(;)g(:::;)g(s)i Fg(\000)h Fo(2;)57 b(0)14 b Fg(\024)g Fn(\022)1183 1275 y Fl(p)p Fi(+)p Fl(q)q Fi(+)p Fl(s)p Fk(\000)p Fi(1)1349 1268 y Fg(\024)g Fo(2)p Fn(\031)r Fo(;)320 b(\(4.5\))0 1378 y(The)16 b(square)h(of)f(a)h(linear)e(elemen)o(t)f(can)i(b)q(e)g (written)g(in)g(the)g(form)426 1488 y Fn(ds)474 1467 y Fi(2)535 1488 y Fo(=)42 b Fn(dr)663 1467 y Fi(2)694 1488 y Fo(+)11 b Fn(r)766 1467 y Fi(2)786 1488 y Fn(d\013)842 1467 y Fi(2)874 1488 y Fo(+)g Fn(r)946 1467 y Fi(2)974 1488 y Fo(sin)1034 1467 y Fi(2)1062 1488 y Fn(\013d\014)1149 1467 y Fi(2)1180 1488 y Fo(+)535 1605 y(+)42 b Fn(r)638 1584 y Fi(2)666 1605 y Fo(cos)731 1584 y Fi(2)759 1605 y Fn(\013)804 1540 y Fl(p)p Fk(\000)p Fi(1)799 1557 y Fj(X)803 1663 y Fl(k)q Fi(=1)879 1542 y Fl(k)q Fk(\000)p Fi(1)880 1557 y Fj(Y)881 1662 y Fl(j)r Fi(=1)952 1605 y Fo(sin)1012 1584 y Fi(2)1040 1605 y Fn(\022)1063 1612 y Fl(j)1081 1605 y Fn(d\022)1130 1584 y Fi(2)1129 1617 y Fl(k)1161 1605 y Fo(+)591 b(\(4.6\))535 1781 y(+)42 b Fn(r)638 1760 y Fi(2)666 1781 y Fo(sin)726 1760 y Fi(2)754 1781 y Fn(\013)9 b Fo(cos)859 1760 y Fi(2)887 1781 y Fn(\014)930 1717 y Fl(q)q Fk(\000)p Fi(1)926 1734 y Fj(X)929 1840 y Fl(k)q Fi(=1)1006 1719 y Fl(k)q Fk(\000)p Fi(1)1006 1734 y Fj(Y)1007 1839 y Fl(j)r Fi(=1)1078 1781 y Fo(sin)1138 1760 y Fi(2)1166 1781 y Fn(\022)1189 1788 y Fl(p)p Fi(+)p Fl(j)1253 1781 y Fn(d\022)1302 1760 y Fi(2)1301 1793 y Fl(p)p Fi(+)p Fl(k)1378 1781 y Fo(+)535 1957 y(+)42 b Fn(r)638 1936 y Fi(2)666 1957 y Fo(sin)726 1936 y Fi(2)754 1957 y Fn(\013)9 b Fo(sin)853 1936 y Fi(2)882 1957 y Fn(\014)925 1895 y Fl(s)p Fk(\000)p Fi(1)920 1910 y Fj(X)924 2016 y Fl(k)q Fi(=1)1000 1895 y Fl(k)q Fk(\000)p Fi(1)1001 1910 y Fj(Y)1002 2015 y Fl(j)r Fi(=1)1073 1957 y Fo(sin)1133 1936 y Fi(2)1161 1957 y Fn(\022)1184 1964 y Fl(p)p Fi(+)p Fl(q)q Fi(+)p Fl(j)1292 1957 y Fn(d\022)1341 1936 y Fi(2)1340 1969 y Fl(p)p Fi(+)p Fl(q)q Fi(+)p Fl(k)1451 1957 y Fn(:)0 2117 y Fo(The)14 b(Laplace)h(equation)f(in)g(trispherical)f(co)q (ordinates)i(is)e(found)i(from)e(\(4.6\))i(and)g(\(2.7\))f(in)g(the)g (form)272 2218 y(1)p 196 2241 177 2 v 196 2286 a Fn(r)219 2272 y Fl(p)p Fi(+)p Fl(q)q Fi(+)p Fl(s)p Fk(\000)p Fi(1)394 2218 y Fn(@)p 382 2241 52 2 v 382 2286 a(@)s(r)439 2252 y(r)462 2232 y Fl(p)p Fi(+)p Fl(q)q Fi(+)p Fl(s)p Fk(\000)p Fi(1)620 2218 y Fn(@)s(u)p 620 2241 57 2 v 622 2286 a(@)s(r)693 2252 y Fo(+)953 2218 y(1)p 747 2241 437 2 v 747 2288 a Fn(r)770 2274 y Fi(2)798 2288 y Fo(cos)863 2274 y Fl(p)p Fk(\000)p Fi(1)936 2288 y Fn(\013)9 b Fo(sin)1036 2268 y Fl(q)q Fi(+)p Fl(s)p Fk(\000)p Fi(1)1152 2288 y Fn(\013)1209 2218 y(@)p 1193 2241 60 2 v 1193 2286 a(@)s(\013)1267 2252 y Fo(cos)1332 2232 y Fl(p)p Fk(\000)p Fi(1)1405 2252 y Fn(\013)g Fo(sin)1505 2232 y Fl(q)q Fi(+)p Fl(s)p Fk(\000)p Fi(1)1621 2252 y Fn(\013)1659 2218 y(@)s(u)p 1657 2241 V 1657 2286 a(@)s(\013)1733 2252 y Fo(+)191 2373 y(+)477 2339 y(1)p 234 2361 512 2 v 234 2409 a Fn(r)257 2394 y Fi(2)285 2409 y Fo(sin)345 2389 y Fi(2)373 2409 y Fn(\013)g Fo(cos)478 2394 y Fl(q)q Fk(\000)p Fl(i)545 2409 y Fn(\014)h Fo(sin)643 2389 y Fl(s)p Fk(\000)p Fi(1)715 2409 y Fn(\014)770 2339 y(@)p 755 2361 59 2 v 755 2407 a(@)s(\014)827 2373 y Fo(cos)893 2352 y Fl(q)q Fk(\000)p Fl(i)959 2373 y Fn(\014)h Fo(sin)1058 2352 y Fl(s)p Fk(\000)p Fi(1)1130 2373 y Fn(\014)1166 2339 y(@)s(u)p 1166 2361 V 1166 2407 a(@)s(\014)1240 2373 y Fo(+)191 2503 y(+)284 2470 y(1)p 234 2492 125 2 v 234 2537 a(cos)299 2523 y Fi(2)327 2537 y Fn(\013)363 2503 y Fo(\()463 2470 y(1)p 387 2492 176 2 v 387 2539 a(sin)447 2519 y Fl(p)p Fk(\000)p Fi(2)520 2539 y Fn(\022)543 2546 y Fi(1)594 2470 y Fn(@)p 573 2492 71 2 v 573 2537 a(@)s(\022)625 2544 y Fi(1)657 2503 y Fo(sin)717 2483 y Fl(p)p Fk(\000)p Fi(2)790 2503 y Fn(\022)813 2510 y Fi(1)845 2470 y Fn(@)s(u)p 838 2492 V 838 2537 a(@)s(\022)890 2544 y Fi(1)925 2503 y Fo(+)827 b(\(4.7\))191 2636 y(+)379 2603 y(1)p 234 2625 315 2 v 234 2672 a(sin)294 2652 y Fi(2)322 2672 y Fn(\022)345 2679 y Fi(1)372 2672 y Fo(sin)432 2652 y Fl(p)p Fk(\000)p Fi(3)506 2672 y Fn(\022)529 2679 y Fi(2)579 2603 y Fn(@)p 558 2625 71 2 v 558 2671 a(@)s(\022)610 2678 y Fi(2)642 2636 y Fo(sin)702 2616 y Fl(p)p Fk(\000)p Fi(3)775 2636 y Fn(\022)798 2643 y Fi(2)830 2603 y Fn(@)s(u)p 823 2625 V 823 2671 a(@)s(\022)875 2678 y Fi(2)910 2636 y Fo(+)11 b Fn(:::)e Fo(+)1299 2603 y(1)p 1064 2625 494 2 v 1064 2672 a(sin)1124 2652 y Fi(2)1152 2672 y Fn(\022)1175 2679 y Fi(1)1203 2672 y Fo(sin)1263 2652 y Fi(2)1291 2672 y Fn(\022)1314 2679 y Fi(2)1333 2672 y Fn(:::)f Fo(sin)1442 2652 y Fi(2)1470 2672 y Fn(\022)1493 2679 y Fl(p)p Fk(\000)p Fi(2)1588 2603 y Fn(@)1617 2585 y Fi(2)1636 2603 y Fn(u)p 1568 2625 117 2 v 1568 2671 a(@)s(\022)1621 2653 y Fi(2)1620 2683 y Fl(p)p Fk(\000)p Fi(1)1689 2636 y Fo(\))j(+)191 2774 y(+)373 2740 y(1)p 234 2763 303 2 v 234 2810 a Fn(r)257 2796 y Fi(2)285 2810 y Fo(sin)345 2790 y Fi(2)373 2810 y Fn(\013)e Fo(cos)478 2796 y Fi(2)506 2810 y Fn(\014)541 2774 y Fo(\()663 2740 y(1)p 565 2763 220 2 v 565 2810 a(sin)625 2790 y Fl(q)q Fk(\000)p Fi(2)697 2810 y Fn(\022)720 2817 y Fl(p)p Fi(+1)839 2740 y Fn(@)p 795 2763 117 2 v 795 2808 a(@)s(\022)847 2815 y Fl(p)p Fi(+1)924 2774 y Fo(sin)984 2754 y Fl(q)q Fk(\000)p Fi(2)1057 2774 y Fn(\022)1080 2781 y Fl(p)p Fi(+1)1179 2740 y Fn(@)s(u)p 1149 2763 V 1149 2808 a(@)s(\022)1201 2815 y Fl(p)p Fi(+1)1282 2774 y Fo(+)i Fn(:::)e Fo(+)p eop %%Page: 11 11 11 10 bop 191 222 a Fo(+)423 188 y(1)p 234 210 404 2 v 234 258 a(sin)294 238 y Fi(2)322 258 y Fn(\022)345 265 y Fl(p)p Fi(+1)418 258 y Fo(sin)477 238 y Fl(q)q Fk(\000)p Fi(3)550 258 y Fn(\022)573 265 y Fl(p)p Fi(+2)691 188 y Fn(@)p 648 210 117 2 v 648 256 a(@)s(\022)700 263 y Fl(p)p Fi(+2)777 222 y Fo(sin)837 201 y Fl(p)p Fk(\000)p Fi(3)910 222 y Fn(\022)933 229 y Fl(p)p Fi(+2)1033 188 y Fn(@)s(u)p 1003 210 V 1003 256 a(@)s(\022)1055 263 y Fl(p)p Fi(+2)1135 222 y Fo(+)191 361 y(+)536 328 y(1)p 234 350 629 2 v 234 397 a(sin)294 377 y Fi(2)322 397 y Fn(\022)345 404 y Fl(p)p Fi(+1)418 397 y Fo(sin)477 377 y Fi(2)506 397 y Fn(\022)529 404 y Fl(p)p Fi(+2)593 397 y Fn(:::)8 b Fo(sin)702 377 y Fi(2)730 397 y Fn(\022)753 404 y Fl(p)p Fi(+)p Fl(q)q Fk(\000)p Fi(2)914 328 y Fn(@)943 310 y Fi(2)962 328 y Fn(u)p 872 350 161 2 v 872 396 a(@)s(\022)925 378 y Fi(2)924 408 y Fl(p)p Fi(+)p Fl(q)q Fk(\000)p Fi(1)1037 361 y Fo(\))k(+)191 499 y(+)370 465 y(1)p 234 488 298 2 v 234 535 a Fn(r)257 521 y Fi(2)285 535 y Fo(sin)345 515 y Fi(2)373 535 y Fn(\013)d Fo(sin)473 515 y Fi(2)501 535 y Fn(\014)536 499 y Fo(\()666 465 y(1)p 560 488 237 2 v 560 535 a(sin)620 515 y Fl(s)p Fk(\000)p Fi(2)691 535 y Fn(\022)714 542 y Fl(p)p Fi(+)p Fl(q)q Fi(1)858 465 y Fn(@)p 806 488 134 2 v 806 533 a(@)s(\022)858 540 y Fl(p)p Fi(+)p Fl(q)q Fi(1)952 499 y Fo(sin)1012 479 y Fl(s)p Fk(\000)p Fi(2)1084 499 y Fn(\022)1107 506 y Fl(p)p Fi(+)p Fl(q)q Fi(+1)1273 465 y Fn(@)s(u)p 1221 488 161 2 v 1221 533 a(@)s(\022)1273 540 y Fl(p)p Fi(+)p Fl(q)q Fi(+1)1398 499 y Fo(+)i Fn(:::)e Fo(+)191 634 y(+)468 600 y(1)p 234 622 492 2 v 234 670 a(sin)294 650 y Fi(2)322 670 y Fn(\022)345 677 y Fl(p)p Fi(+)p Fl(q)q Fi(+1)462 670 y Fo(sin)522 650 y Fl(s)p Fk(\000)p Fi(3)594 670 y Fn(\022)617 677 y Fl(p)p Fi(+)p Fl(q)q Fi(+2)802 600 y Fn(@)p 736 622 161 2 v 736 668 a(@)s(\022)788 675 y Fl(p)p Fi(+)p Fl(q)q Fi(+2)909 634 y Fo(sin)969 613 y Fl(s)p Fk(\000)p Fi(3)1041 634 y Fn(\022)1064 641 y Fl(p)p Fi(+)p Fl(q)q Fi(+2)1230 600 y Fn(@)s(u)p 1178 622 V 1178 668 a(@)s(\022)1230 675 y Fl(p)p Fi(+)p Fl(q)q Fi(+2)1355 634 y Fo(+)191 773 y(+)602 740 y(1)p 234 762 761 2 v 234 809 a(sin)294 789 y Fi(2)322 809 y Fn(\022)345 816 y Fl(p)p Fi(+)p Fl(q)q Fi(+1)462 809 y Fo(sin)522 789 y Fi(2)550 809 y Fn(\022)573 816 y Fl(p)p Fi(+)p Fl(q)q Fi(+2)682 809 y Fn(:::)f Fo(sin)791 789 y Fi(2)819 809 y Fn(\022)842 816 y Fl(p)p Fi(+)p Fl(q)q Fi(+)p Fl(s)p Fk(\000)p Fi(2)1069 740 y Fn(@)1098 722 y Fi(2)1117 740 y Fn(u)p 1005 762 205 2 v 1005 808 a(@)s(\022)1058 790 y Fi(2)1057 820 y Fl(p)p Fi(+)p Fl(q)q Fi(+)p Fl(s)p Fk(\000)p Fi(1)1214 773 y Fo(\))14 b(=)f(0)p Fn(:)73 910 y Fo(As)j(b)q(efore)h(w)o(e)e(are)i(in)o(terested)d(in)i (Eq.\(4.7\))g(a)o(v)o(eraged)g(o)o(v)o(er)g(argumen)o(ts)f Fn(\022)1481 917 y Fi(2)1500 910 y Fn(;)8 b(:::;)g(\022)1609 917 y Fl(p)p Fk(\000)p Fi(1)1672 910 y Fo(,)16 b Fn(\022)1725 917 y Fl(p)p Fi(+2)1790 910 y Fn(;)8 b(:::;)73 970 y(\022)96 977 y Fl(p)p Fi(+)p Fl(q)q Fi(+1)205 970 y Fo(,)k(...,)f Fn(\022)321 977 y Fl(p)p Fi(+)p Fl(q)q Fi(+)p Fl(s)p Fk(\000)p Fi(1)474 970 y Fo(.)20 b(Denoting)12 b Fn(\022)737 977 y Fi(1)770 970 y Fo(=)i Fn(\022)q Fo(,)f Fn(\022)896 977 y Fl(p)p Fi(+1)974 970 y Fo(=)h Fn( )r Fo(,)e(w)o(e)f(can)h(write)f (the)h(in)o(terp)q(olated)f(equation)0 1030 y(in)16 b(the)g(form)518 1111 y(1)p 442 1133 177 2 v 442 1179 a Fn(r)465 1164 y Fl(p)p Fi(+)p Fl(q)q Fi(+)p Fl(s)p Fk(\000)p Fi(1)640 1111 y Fn(@)p 629 1133 52 2 v 629 1179 a(@)s(r)686 1144 y(r)709 1124 y Fl(p)p Fi(+)p Fl(q)q Fi(+)p Fl(s)p Fk(\000)p Fi(1)867 1111 y Fn(@)s(u)p 867 1133 57 2 v 869 1179 a(@)s(r)939 1144 y Fo(+)437 1263 y(+)687 1230 y(1)p 480 1252 437 2 v 480 1299 a Fn(r)503 1285 y Fi(2)532 1299 y Fo(cos)597 1285 y Fl(p)p Fk(\000)p Fi(1)670 1299 y Fn(\013)9 b Fo(sin)770 1279 y Fl(q)q Fi(+)p Fl(s)p Fk(\000)p Fi(1)886 1299 y Fn(\013)943 1230 y(@)p 927 1252 60 2 v 927 1298 a(@)s(\013)1000 1263 y Fo(cos)1066 1243 y Fl(p)p Fk(\000)p Fi(1)1139 1263 y Fn(\013)f Fo(sin)1238 1243 y Fl(q)q Fi(+)p Fl(s)p Fk(\000)p Fi(1)1355 1263 y Fn(\013)1393 1230 y(@)s(u)p 1391 1252 V 1391 1298 a(@)s(\013)1467 1263 y Fo(+)437 1384 y(+)724 1350 y(1)p 480 1373 512 2 v 480 1420 a Fn(r)503 1406 y Fi(2)532 1420 y Fo(sin)592 1400 y Fi(2)620 1420 y Fn(\013)g Fo(cos)725 1406 y Fl(q)q Fk(\000)p Fl(i)791 1420 y Fn(\014)j Fo(sin)890 1400 y Fl(s)p Fk(\000)p Fi(1)962 1420 y Fn(\014)1017 1350 y(@)p 1002 1373 59 2 v 1002 1418 a(@)s(\014)1074 1384 y Fo(cos)1139 1364 y Fl(q)q Fk(\000)p Fl(i)1206 1384 y Fn(\014)g Fo(sin)1304 1364 y Fl(s)p Fk(\000)p Fi(1)1376 1384 y Fn(\014)1413 1350 y(@)s(u)p 1412 1373 V 1412 1418 a(@)s(\014)1486 1384 y Fo(+)266 b(\(4.8\))437 1515 y(+)556 1481 y(1)p 480 1503 177 2 v 480 1549 a Fn(r)503 1534 y Fi(2)532 1549 y Fo(cos)597 1534 y Fi(2)625 1549 y Fn(\013)733 1481 y Fo(1)p 666 1503 158 2 v 666 1550 a(sin)726 1531 y Fl(p)p Fk(\000)p Fi(2)799 1550 y Fn(\022)846 1481 y(@)p 833 1503 53 2 v 833 1549 a(@)s(\022)899 1515 y Fo(sin)959 1494 y Fl(p)p Fk(\000)p Fi(2)1032 1515 y Fn(\022)1061 1481 y(@)s(u)p 1061 1503 57 2 v 1063 1549 a(@)s(\022)1134 1515 y Fo(+)437 1635 y(+)619 1602 y(1)p 480 1624 303 2 v 480 1671 a Fn(r)503 1657 y Fi(2)532 1671 y Fo(sin)592 1651 y Fi(2)620 1671 y Fn(\013)8 b Fo(cos)725 1657 y Fi(2)753 1671 y Fn(\014)864 1602 y Fo(1)p 793 1624 166 2 v 793 1671 a(sin)853 1651 y Fl(q)q Fk(\000)p Fi(2)925 1671 y Fn( )985 1602 y(@)p 969 1624 62 2 v 969 1670 a(@)s( )1044 1635 y Fo(sin)1104 1615 y Fl(q)q Fk(\000)p Fi(2)1176 1635 y Fn( )1217 1602 y(@)s(u)p 1215 1624 V 1215 1670 a(@)s( )1295 1635 y Fo(=)14 b(0)p Fn(:)0 1761 y Fo(In)i(Cartesian)h(co) q(ordinates)157 1863 y Fn(x)185 1870 y Fi(1)246 1863 y Fo(=)42 b Fn(r)9 b Fo(cos)g Fn(\013)f Fo(cos)h Fn(\022)q(;)57 b(x)667 1870 y Fi(2)700 1863 y Fo(=)14 b Fn(r)9 b Fo(cos)g Fn(\013)g Fo(sin)f Fn(\022)q(;)787 b Fo(\(4.9\))112 1936 y Fn(x)140 1943 y Fl(p)p Fi(+1)246 1936 y Fo(=)42 b Fn(r)9 b Fo(sin)f Fn(\013)h Fo(cos)g Fn(\014)h Fo(cos)f Fn( )r(;)56 b(x)783 1943 y Fl(p)p Fi(+2)861 1936 y Fo(=)14 b Fn(r)c Fo(sin)e Fn(\013)h Fo(cos)f Fn(\014)j Fo(sin)d Fn( )r(;)56 b(x)1365 1943 y Fl(p)p Fi(+)p Fl(q)q Fi(+1)1488 1936 y Fo(=)14 b Fn(r)9 b Fo(sin)g Fn(\013)f Fo(sin)g Fn(\014)0 2038 y Fo(Eq.)21 b(\(4.8\))c(rearranges)g(to)f(the)g(form)519 2123 y Fi(2)492 2138 y Fj(X)496 2244 y Fl(k)q Fi(=1)564 2185 y Fo(\()589 2152 y Fn(@)618 2133 y Fi(2)637 2152 y Fn(u)p 588 2174 78 2 v 588 2219 a(@)s(x)645 2202 y Fi(2)645 2233 y Fl(k)681 2185 y Fo(+)759 2152 y Fn(@)788 2133 y Fi(2)807 2152 y Fn(u)p 735 2174 123 2 v 735 2219 a(@)s(x)792 2202 y Fi(2)792 2233 y Fl(p)p Fi(+)p Fl(k)863 2185 y Fo(\))11 b(+)992 2152 y Fn(@)1021 2133 y Fi(2)1040 2152 y Fn(u)p 947 2174 166 2 v 947 2219 a(@)s(x)1004 2202 y Fi(2)1004 2232 y Fl(p)p Fi(+)p Fl(q)q Fi(+1)1128 2185 y Fo(+)1182 2152 y(\()p Fn(p)h Fg(\000)f Fo(2\))p 1182 2174 148 2 v 1233 2219 a Fn(x)1261 2226 y Fi(2)1350 2152 y Fn(@)s(u)p 1340 2174 76 2 v 1340 2219 a(@)s(x)1397 2226 y Fi(2)1432 2185 y Fo(+)295 b(\(4.10\))492 2337 y(+)535 2303 y(\()p Fn(q)12 b Fg(\000)f Fo(2\))p 535 2325 147 2 v 562 2371 a Fn(x)590 2378 y Fl(p)p Fi(+2)724 2303 y Fn(@)s(u)p 691 2325 122 2 v 691 2371 a(@)s(x)748 2378 y Fl(p)p Fi(+2)828 2337 y Fo(+)882 2303 y(\()p Fn(s)g Fg(\000)g Fo(1\))p 882 2325 147 2 v 887 2371 a Fn(x)915 2378 y Fl(p)p Fi(+)p Fl(q)q Fi(+1)1093 2303 y Fn(@)s(u)p 1039 2325 166 2 v 1039 2371 a(@)s(x)1096 2378 y Fl(p)p Fi(+)p Fl(q)q Fi(+1)1223 2337 y Fo(=)j(0)p Fn(:)0 2468 y Fo(It)e(is)g(seen)g(from)f(\(4.10\))j(that)e(Eq.\(4.10\))h(\(and)g (Eq.\(4.8\))f(resp)q(ectiv)o(ely\))e(p)q(ermits)h(only)h(rotations)h (and)0 2528 y(parallel)g(transfers)i(in)e(\()p Fn(x)470 2535 y Fi(1)490 2528 y Fn(;)8 b(x)540 2535 y Fl(p)p Fi(+1)604 2528 y Fo(\)-plane)14 b(for)h(arbitrary)f(v)m(alues)g(of)g Fn(p;)8 b(q)16 b Fo(and)f Fn(s)p Fo(.)20 b(This)14 b(rotation)h(yields) 0 2589 y(only)i(an)g(asymmetric)c(addition)k(form)o(ula)e(as)i(in)g (the)f(case)h(of)g(bispherical)f(co)q(ordinates.)23 b(Ho)o(w)o(ev)o (er,)0 2649 y(w)o(e)15 b(can)g(obtain)h(the)f(symmetric)d(form)o(ula)i (in)h(trispherical)f(case)h(using)h(paratactical)f(displacemen)o(t)0 2709 y(in)h(the)g(subspace)h(\()p Fn(x)392 2716 y Fi(1)411 2709 y Fn(;)8 b(x)461 2716 y Fi(2)480 2709 y Fn(;)g(x)530 2716 y Fl(p)p Fi(+1)595 2709 y Fn(;)g(x)645 2716 y Fl(p)p Fi(+2)709 2709 y Fo(\):)173 2811 y Fn(x)201 2791 y Fk(0)201 2823 y Fl(k)236 2811 y Fo(=)14 b Fn(x)316 2818 y Fl(k)345 2811 y Fo(cos)9 b Fn(\034)17 b Fg(\000)11 b Fn(x)535 2818 y Fl(p)p Fi(+)p Fl(k)609 2811 y Fo(sin)d Fn(\034)s(;)57 b(x)800 2791 y Fk(0)800 2823 y Fl(p)p Fi(+)p Fl(k)880 2811 y Fo(=)14 b Fn(x)960 2818 y Fl(k)989 2811 y Fo(sin)8 b Fn(\034)17 b Fo(+)11 b Fn(x)1172 2818 y Fl(p)p Fi(+)p Fl(k)1247 2811 y Fo(cos)d Fn(\034)s(;)57 b(k)16 b Fo(=)e(1)p Fn(;)8 b Fo(2)p Fn(:)173 b Fo(\(4.11\))p eop %%Page: 12 12 12 11 bop 73 203 a Fo(The)16 b(axis)h Fn(x)301 210 y Fl(p)p Fi(+)p Fl(q)q Fi(+1)426 203 y Fo(remains)e(motionless:)768 308 y Fn(x)796 287 y Fk(0)796 320 y Fl(p)p Fi(+)p Fl(q)q Fi(+1)919 308 y Fo(=)f Fn(x)999 315 y Fl(p)p Fi(+)p Fl(q)q Fi(+1)1108 308 y Fn(:)643 b Fo(\(4.12\))73 412 y(Let)12 b(us)g(in)o(tro)q(duce)g(the)f(trispherical)g(co)q(ordinates)i (\(4.1\)-\(4.5\))f(in)g(the)f(initial)g(co)q(ordinate)h(system)0 473 y(\()p Fn(x)47 480 y Fi(1)66 473 y Fn(;)c(:::;)g(x)180 480 y Fl(p)p Fi(+)p Fl(q)q Fi(+1)288 473 y Fo(\))20 b(and)h(in)f(the)g (transformed)f(system)g(\()p Fn(x)1064 454 y Fk(0)1064 485 y Fi(1)1083 473 y Fn(;)8 b(:::;)g(x)1197 454 y Fk(0)1197 485 y Fl(p)p Fi(+)p Fl(q)q Fi(+1)1305 473 y Fo(\).)33 b(Then)20 b(from)f(the)h(\014rst)h(t)o(w)o(o)0 533 y(form)o(ulae)15 b(\(4.11\))h(w)o(e)g(obtain)h(the)f(follo)o(wing)g(relation)51 637 y(cos)116 617 y Fi(2)144 637 y Fn(\013)175 617 y Fk(0)201 637 y Fo(=)e(cos)318 617 y Fi(2)346 637 y Fn(\013)9 b Fo(cos)451 617 y Fi(2)479 637 y Fn(\034)17 b Fo(+)11 b(sin)626 617 y Fi(2)654 637 y Fn(\013)e Fo(sin)753 617 y Fi(2)781 637 y Fn(\034)14 b Fo(cos)882 617 y Fi(2)910 637 y Fn(\014)f Fg(\000)e Fo(\(1)p Fn(=)p Fo(2\))d(sin)i(2)p Fn(\013)f Fo(sin)f(2)p Fn(\034)14 b Fo(cos)9 b Fn(\014)h Fo(cos)q(\()p Fn(\022)i Fg(\000)f Fn( )r Fo(\))50 b(\(4.13\))0 742 y(i.e)15 b(the)h(symme)o(tric)d(transformation)j(whic)o(h)f(w)o(as) h(included)f(in)h(\(1.1\).)21 b(The)16 b(paratactical)g(displace-)0 802 y(men)o(t)e(\(4.11\))k(is)e(p)q(ermissible)e(for)j(Eqs.)22 b(\(4.8\),)16 b(\(4.10\))h(only)f(if)g Fn(p)f Fo(=)f Fn(q)h Fo(=)g(2,)h(and)h(therefore)f(the)g(only)0 862 y(parameter)f(for)i(in)o(terp)q(olation)e(is)i(parameter)e Fn(s)p Fo(.)73 922 y(Let)24 b(us)f(construct)h(the)f(singular)g (solution)h(of)g(\(4.8\))f(\()p Fn(p)j Fo(=)g Fn(q)i Fo(=)d(2\))f(with)f(a)h(singularit)o(y)e(at)0 982 y Fn(r)i Fo(=)e Fn(r)128 989 y Fi(0)148 982 y Fn(;)8 b(\013)22 b Fo(=)g(0.)37 b(It)20 b(is)h(seen)g(from)f(\(4.9\))i(that)g(the)e (line)h(of)g(singularities)g(is)g(the)f(circumference)0 1043 y Fn(x)28 1050 y Fi(10)82 1043 y Fo(=)c Fn(r)158 1050 y Fi(0)186 1043 y Fo(cos)9 b Fn(\022)284 1024 y Fk(\003)304 1043 y Fn(;)f(x)354 1050 y Fi(20)407 1043 y Fo(=)17 b Fn(r)484 1050 y Fi(0)512 1043 y Fo(sin)8 b Fn(\022)604 1024 y Fk(\003)642 1043 y 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1388 y Fg(\000)g Fn(x)811 1395 y Fi(20)848 1388 y Fo(\))867 1367 y Fi(2)898 1388 y Fo(+)g Fn(x)975 1367 y Fi(2)975 1400 y Fl(p)p Fi(+1)1050 1388 y Fo(+)g Fn(x)1127 1367 y Fi(2)1127 1400 y Fl(p)p Fi(+2)1203 1388 y Fo(+)g Fn(x)1280 1367 y Fi(2)1280 1400 y Fl(p)p Fi(+)p Fl(q)q Fi(+1)1389 1388 y Fo(])1403 1367 y Fk(\000)p Fi(\()p Fl(s)p Fi(+2\))p Fl(=)p Fi(2)1570 1388 y Fo(=)351 1469 y(=)j(\()p Fn(r)445 1448 y Fi(2)476 1469 y Fo(+)d Fn(r)548 1448 y Fi(2)547 1481 y(0)579 1469 y Fg(\000)g Fo(2)p Fn(r)q(r)698 1476 y Fi(0)726 1469 y Fo(cos)q(\()p Fn(\022)h Fg(\000)f Fn(\022)920 1448 y Fk(\003)939 1469 y Fo(\)\))977 1448 y Fk(\000)p Fi(\()p Fl(s)p Fi(+2\))p Fl(=)p Fi(2)1131 1469 y Fn(:)620 b Fo(\(4.14\))73 1573 y(Constructing)24 b(the)f(same)g(solution)g(in)g(the)g(separated)h (trispherical)e(co)q(ordinates)i(giv)o(es)f(the)0 1633 y(iden)o(tit)o(y)319 1738 y(\(1)12 b(+)f Fn(t)441 1717 y Fi(2)471 1738 y Fg(\000)g Fo(2)p Fn(t)d Fo(cos)h Fn(\013)f Fo(cos)h Fn(')p Fo(\))809 1717 y Fk(\000)p Fi(\()p 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y(No)o(w)k(w)o(e)f(can)g(rep)q(eat)h(our)g(general)f(pro)q (cedure,)h(i.e.,)f(construct)g(the)h(solution)f(of)h(\(5.3\))g(with)0 2026 y(singularit)o(y)16 b(at)h Fn(r)f Fo(=)f Fn(r)415 2033 y Fi(0)434 2026 y Fn(;)8 b(\013)15 b Fo(=)g(0)p Fn(;)i Fo(sub)s(ject)f(this)g(solution)h(to)h(transformation)e(\(5.2\)) h(and)h(equate)e(the)0 2086 y(solutions)h(in)f(v)m(ariables)g(\()p Fn(r)o(;)8 b(\013;)g(\014)s(;)g(!)r Fo(\))15 b(and)i(\()p Fn(r)o(;)8 b(\013)885 2068 y Fk(0)896 2086 y Fn(;)g(\014)949 2068 y Fk(0)960 2086 y Fn(;)g(!)1014 2068 y Fk(0)1026 2086 y Fo(\))16 b(to)h(obtain)f(the)g(addition)h(form)o(ula)e(\(1.1\).) 73 2146 y(The)c(same)f(tec)o(hnique)g(can)h(b)q(e)g(used)g(in)g(the)g (case)g(of)g(h)o(yp)q(ergeometric)e(function.)19 b(Let)12 b(us)f(consider)0 2206 y(pseudo-Euclidean)18 b(space)g Fn(E)558 2213 y Fi(2)p Fl(p)p Fi(+)p Fl(s)p Fk(\000)p Fi(1)p Fl(;)p Fi(1)729 2206 y Fo(instead)g(of)g(the)f(Euclidean)g (space)h Fn(E)1436 2213 y Fi(2)p Fl(p)p Fi(+)p Fl(s)1517 2206 y Fo(.)25 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Fl(p)p Fk(\000)p Fi(1)676 2836 y Fn(\013)9 b Fo(sinh)803 2815 y Fl(q)q Fi(+)p Fl(s)p Fk(\000)p Fi(1)919 2836 y Fn(\013)976 2765 y(@)p 960 2787 60 2 v 960 2833 a(@)s(\013)1033 2799 y Fo(cosh)1126 2778 y Fl(p)p Fk(\000)p Fi(1)1199 2799 y Fn(\013)g Fo(sinh)1326 2778 y Fl(q)q Fi(+)p Fl(s)p Fk(\000)p Fi(1)1442 2799 y Fn(\013)1480 2765 y(@)s(u)p 1478 2787 V 1478 2833 a(@)s(\013)1543 2799 y(:)p eop %%Page: 16 16 16 15 bop 73 203 a Fo(Limiting)13 b(ourselv)o(es)h(to)i(the)e(case)h Fn(p)g Fo(=)e Fn(q)r(;)h Fo(in)o(tro)q(ducing)h(the)g(v)m(ariable)f Fn(!)j Fo(b)o(y)e(the)f(relations)h(\(5.5\),)0 264 y(\(5.7\))j(and)h (separating)g(co)q(ordinate)f Fn(r)i Fo(in)d(the)h(resulting)g (equation)f(w)o(e)h(come)e(to)j(the)f(equation)g(for)0 324 y(the)e(spherical)g(functions)g(\(on)g(h)o(yp)q(erb)q(oloid\))h(in) f(the)g(form)650 408 y(1)p 442 430 441 2 v 442 479 a(cosh)534 457 y Fl(p)p Fk(\000)p Fi(1)607 479 y Fn(\013)9 b Fo(sinh)734 457 y Fl(p)p Fi(+)p Fl(s)p Fk(\000)p Fi(1)851 479 y Fn(\013)908 408 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Fn(\013)9 b Fo(sinh)404 2591 y Fl(p)p Fi(+)p Fl(s)p Fk(\000)p Fi(1)521 2613 y Fn(\013)578 2542 y(d)p 562 2564 57 2 v 562 2610 a(d\013)632 2575 y Fo(cosh)724 2554 y Fl(p)p Fk(\000)p Fi(1)798 2575 y Fn(\013)f Fo(sinh)924 2554 y Fl(p)p Fi(+)p Fl(s)p Fk(\000)p Fi(1)1041 2575 y Fn(\013)1079 2542 y(du)p 1077 2564 V 1077 2610 a(d\013)1150 2575 y Fo(+)j Fn(\027)s Fo(\(2)p Fn(p)h Fo(+)f Fn(s)g Fo(+)g Fn(\027)k Fg(\000)c Fo(2\))p Fn(u)j Fo(=)f(0)p Fn(:)0 2691 y Fo(The)i(alternativ)o(e)f(w)o(a)o(y)h (of)h(deriving)e(the)h(addition)h(form)o(ula)e(for)h(Jacobi)h(function) f Fn(P)1588 2673 y Fm({)q Fl(;\025)1581 2703 y(\027)1645 2691 y Fo(\()p Fn(x)p Fo(\))g(consists)0 2751 y(in)f(using)g(Mellin)f (in)o(tegral)g(transform)h(in)g Fn(r)i Fo(for)e(presen)o(ting)g(the)g (singular)g(solution)h(of)f(Eq.\(5.3\))g(and)0 2811 y(transforming)i (of)g(the)g(mo)q(di\014ed)g(singular)g(solution)h(b)o(y)e(rotation)i (\(5.2\).)p eop %%Page: 17 17 17 16 bop 0 203 a Fp(Note:)0 313 y Fo(This)13 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