234 



PROFESSOR A. E. H. LOVE ON THE 



58. Let F(0, <) denote the function to be expanded. The coefficients are expressed 

 by equations of the type 



p ( " <ty f (16 (cos < sin e) 2 sin 6 = \ d<f> f d0F(0, <f>) (cos <f> sin 6} sin 0. . (11 9) 



J o J o J o J o 



The factors multiplying the coefficients p, &c. in the left-hand members are the 

 integrals of the squares of the several harmonics over the surface of a unit sphere. 

 The integrals in the right-hand members are the integrals, over the surface of the 

 same sphere, of the product of the function to be expanded and the corresponding 

 harmonics. The values of the integrated squares multiplying p, &c., are recorded in 



the following table : 



Coefficient. 



p,q,r 



a, , 7' 



d,e 



f,9 



Value of integrated 

 square. 



1 



Reciprocals. 



Since the ratios only are relevant, the integrals in the right-hand members of such 

 equations as (119) are to be multiplied by the numbers in brackets in the third 

 column. 



59. To evaluate integrals of the type in the right-hand member of (119), when the 

 value of F (0, <f>) is given by the table of 56, or any similar table, we treat the 

 integral as a double sum, e.g., 



71 



' 



30 



36 



then we have to evaluate such a double sum as 



v v V i H7r mir \ 

 2, j I , 



m=o B=I '^oo 30 y 



u ~ 



