432 Mr. J. H. Jeans on the 



We write L = M = N=0 in equations (5), (6) and (7); 

 replace d 2 /dt 2 by — jo 2 , and substitute for u, v, w from scheme 

 (1). The resulting equations must then be identities for all 

 values of r, 0, and <£. 



To save printing, we shall write 



W= ^ 



Jm 

 n = 



and 



d 2 V, 



dr* ' dr* ' 



Jn = pF^ H- 0-G» , 



SJT = ^ dG? 

 o> ^ dr dr 



In this abbreviated notation, the transformed equations (5), 

 (6) and (7) are found to be 



^ss^rpro*)^^=22(^+«:w)pr(^)^ . . (8) 



opV 22 /8TPro*)^=2S JT ^ Jg { sin £^M } ***, (9) 



oy V sin 2 022 tJPJ (^) **"* = 22 JT«mPT (/*) ^ m(p . . . (10) 



Equation (8) is expanded on both sides in zonal, tesseral, 

 and sectorial harmonics, and is true for all values of and </>. 

 We may therefore equate the coefficients of the various har- 

 monics, and obtain 



cr/< = &? +«»W (11) 



or 



In equations (9) and (10) we may begin by equating the 

 coefficients of e im<p ; multiply the equation obtained in this 

 manner from (10) by im/sin 2 6 and add the result to the 

 equation obtained by equating coefficients of e im Q in (9). The 

 result is 



The right-hand member is equal to 



-s»(n+i)jrpro*), 



and we are now able to equate coefficients of P»(/tt). The 



