

SQUARES OF ELLIPSOIDAL SURFACE HARMONIC FUNCTIONS. 11 



since it is sometimes tedious, and it merely involves the substitution in the formula 

 of the values of A Q , A^ An, &c. 



We will now take the several harmonics successively. 



HARMONIC OF THE ORDELI ZERO. 



This harmonic is simply unity, so that A l} = 1 and all other .1's vanish. The 

 formula is 



This is obviously right since the integral is |^"/T, of which this is the known value. 



HARMONICS OF TL-IK FIRST OKUEK. 



Here we have all the .1's zero excepting .1,, ;md .1,. and \vlien the functions have 

 the proper symmetrical forms, we have from (i!), 



r,/cos\ _ 47rA," ) Cos B cos y ,- , ., i e .. 2 _L 9 / 2 >:\ i i -\ / . oh 



' Isin/ " "sin ;J {3~~ 



(I) Tlie Zonal Harmonic. 

 I define this thus, 



1 . . . ,,. 



= ( I - /c' : cos- r/.)-' = // . (K- + '- sin 3 </;)', J 



1 



where J = , y = I. 



K 



On squaring Jj^ i (p), it is clear that 



Whence I find 



T / \ 4?r/; 3 cos B cos y /. 2 



Since with definition (-l)f~g-K = 1, 



A __ -l^ 3 cos )8 cos y 



In " Harmonics " this harmonic is defined by 



-^cos2^) . . . .'(6). 



