CLASS OF DEFINITE INTEGRALS AND INFINITE SERIES. 



187 



26. The preceding examples will be sufficient to illustrate the general method. I will 

 remark in conclusion that the process of integration applied to the equations (ll), (47), and 

 (58) leads very readily to the complete integral in finite terms of the equation 



2 w ( . i(i + 1)1 



d*y 

 d 



(67) 



where i is an integer, which without loss of generality may be supposed positive, 

 under which the integral immediately comes out is 



The form 



V 



a#* \\ ~ i ( * + 1} + (i " x) * (i + 1} (i + 2) 



1 . %qx 



_ , i (i + 1) (i 



4 1, 



1 .2(2qxf 

 1) i (i + l) (i + 2) 



■1. 



,2qai l.»(9qaf 



where each series will evidently contain i + 1 terms. It is well known that (67) is a general 

 integrable form which includes as a particular case the equation which occurs in the theory of 

 the figure of the earth, for q in (67) is any quantity real or imaginary, and therefore the equa- 

 tion formed from (67) by writing + q*y for — q 2 y may be supposed included in the form (67). 



It may be remarked that the differential equations discussed in this paper can all be reduced 

 to particular cases of the equation obtained by replacing i (i + 1) in (67) by a general constant. 

 By taking gr$, where g is any constant, for the independent variable in place of n in the dif- 

 ferential equations which U, V in the first example satisfy, these equations are reduced to the 

 form 



d*y 2a dy lb 



and (47), (58) are in this form already 

 equation to the form required. 



2a dy lb \ 



Putting now y = x~"x, we shall reduce the last 



G. G. STOKES. 



Pembroke College, 

 Feb. 4, 1850. 



