436 Mr. R. Moon on the Equation of Laplace's Coefficients. 



(6\ 

 tan 2/ 



,/ . A^k A dk. - . ' A \ 



y l = Msmu--jQ2 + cos " Ja~ +»• w+lsin^. Ah 



7 2 =|(sinfl * +cos0--7^ + n.n + l sin . A h 



&c. 

 whence we have 

 A =c, 



&c, 



A 1 =-j , 7 1 ^=-i(sin^ +w . w+ ij s i n ^.A.^ \ 

 A 2 = — Jy 2 ^= — i^sin 0-^> + w . w+ljsin 0.A 1 .d9'ji 



A 3 = — §y 8 d0= — i(sin -j^ + n.n+l^sinO .A^.dO \ 



&c. &c. ; 



or, substituting for A and performing the integrations, 



r L (2) 



. 1 f n.n + l a . "\ 



A 1 = - -j c. — j .cos^ + Cj >, 



A 2= l 2 {e/-^±I^^ 



n.n+l(n.n + l-1.2)(n.w + l-2.3) cos3 ^ 

 1.2.3 



A 8 - 2^ 



+ c i 



n.n + l{n.n+l~ 1 .2) 

 1.2 



cos 2 (9 



L + (c.n. w+l(n.w+l — 1.2) +c 2 .n.n+l)cos6 + c 3 



&c. &c. 



If the condition be imposed that the series shall terminate, it 

 maybe satisfied by means of the constants introduced in the sub- 

 sidiary integrations. 



* Instead of taking for the base of the arbitrary function the above value 

 of u, we may take e ±u for the base where u has the above value, the form 

 so obtained being identical with that given by Professor Donkin. Its adop- 

 tion, however, adds materially to the complexity of the result. 



