Mr. R. Moon on the Equation of Laplace's Coefficients. 435 



or by a pair of such series; where ^ is arbitrary, u is determined 

 by the equation 



_ ^ du I ~du du m du^ 

 dx ' doc ay dy 



A = e-/W», 



A 2 = -e-fW^dx . 7 s e/** 



&c. &c, 



where 



,, 6?<2? 2 cMy dy 1 dx dy 



„rf 2 A , _ </ 2 A , „,rf*A , „rfA „«?A , TT . 



71 2R*- + S^ 



dx dy 



aar Kfli/ dy z ax dy 



«# ay 

 &c. &c. 



The equation of Laplace's coefficients may readily be inte- 

 grated by this general method, some of the peculiarities of which 

 are well illustrated by its application to that equation, of which 

 it probably offers the simplest solution which is obtainable. 



Retaining the original spherical coordinates, the equation may 

 be written 



d C2 co . . a „ d 2 co . . n „dco 



= -j-jj-g- + sin 2 6 -77j2 + sin 6 cos 6 -^ + n . n -f- 1 . sin 2 . <o ; 



and putting 6 and <f> for x and y respectively in the preceding 

 formulae, we shall find 



* In the paper above referred to, I have stated "that no constants are 

 to be introduced " in effecting the integrations here indicated. This is true 

 so long as in (1) we have U = ; but where this does not hold, the omission 

 of the constants would greatly curtail the generality of the result, as a 

 glance at the mode in which each of the quantities A a , A 2 , &c. is derived 

 from its immediate predecessor will at once show. 



