Mr. R. Moon on the Equation of Laplace's Coefficients, 437 



To effect this, let A m be the coefficient of the last term of the 

 series for co ; then from what has preceded it is evident that we 

 may assume 



A m =a m cos 0| w + «m-i cos^p- 1 + a m . 2 cos 6\ m ~ 2 + &c, 



where a mi a m _ ly &c. are constants ; and since by hypothesis 

 A^^ssO, the formula? (2) give us 



= sin d 4^ +n . n"+l S sin * ' A - ^* 



The substitution in this equation of the above expression for A m 

 gives 



= ma m+1 cosd\ m+] +m—la m -iCOs6\ m + m — 2a m - 2 co$d\ m ~ 1 



+ m-3tf m _ 3 cos0| m ~ 2 +&c. 



-m^coi^h- 1 -^^!^.! cos~0| m_2 + &c. 



cos6f +1 cosC T cos6> l' 



-n.n + la m -^-^- -n.n + la^— w - w + lfl m-a^j 



*r 



cos Jm ~ 2 



— ft. w + l« m _ 3 4r- + &C *J 



m— 2 



whence we derive the equations 



0=(m- 1^+iV, 



- / , ft. W-hlV 



\ m ' 



0=(m-2- n ' n+ l l )a m -2-ma m} 

 \ m — 1 / 



_ / Q ft.ft-fl\ r 



\ m— 2 / 



&c. &c, 



and, ultimately, the following, viz. : — 

 m = ft, 



«m-i = 0, 



n . ft — 1 



#m— 2 — 



« w _ 3 =0, 



ft . ft— 1 ft — 2 ft — 3 



(ft.ft + 1 — rc — 1 n— 2)(ft .ft-fl — ft — 3/i— 4) 

 &c. &c, 



