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



(v) 



dh dh 



u = Cl U+c 2 V'; 



(vi) 



ihh d>? + *m- u = ' 



u = cJ+cJJ'; 

 (vii) 



«»'£+ 4 (*'-*>£ +».-<* 



?* = c 1 TF+c 2 Tf'. 



§ 18. The differential equations may be very easily obtained in 

 the form given in the last section by means of the differential 

 formula? : 



which sfive 



Definitions of K, E, I, &c. as integrals, §§ 19, 20. 



§ 19. Denoting by X the expression 



V{(1-<>(1-&V)}, 



the definitions of K, E, I, &c. as integrals are 



f 1 1 

 (i) K= J j.dx, 



(ii) E= J — -^ — dx, 



(iii) 1 = - -^ dx, 



Jo -A 



