1884.] K', E', J\ G' in 'powers of the modulus. 197 



(vi) 



u = CjV + c 2 U'; 

 (vii) 



k (1 - F) ^ + (1 - 3F) % + Sku = 0, 



aAr dk 



u = c 1 W+c 2 W, 

 where c x and c a are arbitrary constants. 



§ 17. These differential equations assume a more elegant form 

 when the independent variable is taken to be k 2 instead of k. 



Putting h = k\ li=k'\ 



so that h + li = 1 , 



the differential equations are: 



(i) 



dh v J dh 

 u — cj£+ c 2 K'; 



(ii) 



4M' -^ + 4A' i? + m = 0, 



a/i em 



(iii) 

 4/i/i -77-0 — 4/i -77- -f w = 0, 



