(v) U=\ ^ ** dx, 



1884.] K', E', J', Gr in powers of the modulus. 199 



(iv) G = I y dx, 



-I 

 o X 



(vi) V=J ^ -^ dx, 



( v ii) TT= / ^-±^dx, 

 the track of integration in each case being real. 



§ 20. If the track be not real, but be the same for all 

 the integrals, the corresponding simultaneous values of the inte- 

 grals are 



(i) (2m+l)K+2niK', 



(ii) (2m + l)E-2nir, 



(iii) (2m + 1) J - 2niE\ 



(iv) (2m + \)G-2niG, 



(v) {2m + l)U-2niV, 



(vi) (2m + l)V-2niU', 



(vii) {2m + l)W-2niW. 



Note on the Differential Equations, § 21. 



§ 21. Denning K, E, I, &c. as in § 19 by means of the inte- 

 grals, we may shew, by differentiation and transformation of these 

 integrals, that they satisfy the differential equations (i), ... (vii). 

 The validity of the processes is in no way dependent upon the 

 reality of the track of integration, though it must be the same in 

 the case of each of the integrals used in verifying the same 

 differential equation. The seven differential equations are there- 

 fore satisfied respectively by the seven values of the integrals 

 given in the last section. Thus the differential equation satisfied 

 by K is satisfied by K' also, that satisfied by E is satisfied by /' 

 also, and so on. This explanation of the fact that K and K', E 

 and /', &c. satisfy the same differential equations is well known : 

 it is only referred to here in order to notice that the same ex- 

 planation applies in the case of each of the seven differential 

 equations. 



