196 Mr Glaisher, On the developments of [Nov. 24, 



Differential equations satisfied by K and K', E and I', &c, 



§§ 16-18. 



§ 16. It is well known that K and K' satisfy the same differ- 

 ential equation of the second order, viz. 



and it was in fact by means of this differential equation that 

 Legendre obtained the series for K' in terms of k. It is well 

 known also that the quantities E and K' — E' satisfy the same 

 differential equation; whence also E' and K — E satisfy the same 

 differential equation. It can be shewn that the same differential 

 equations are satisfied by G and 0', by U and V, by V and U', 

 and by W and W. These seven differential equations and their 

 complete integrals are as follows : 



(i) 

 k(l-k*)^ 2 + (l-3k*)^-ku = 0, 



u = c t K + c 2 K'; 

 (ii) 



u = c 1 E + c 2 I'; 

 (iii) 

 k(l-k^-(l + k>)f k + ku=-0, 

 u = c 1 I+c z E'; 



(iv) 



u = Cj6r + c 2 G'; 



(v) 

 *(l-^-(8-^-*u = 0, 



u = c 1 t/ r +c 2 F'; 



