282 Mr. W. H. L. Russell on [Feb. 2, 



Hence the differential equation becomes 



(m+x) g _ + + v + («+ (3 + y)(m +#)} g 



+ + y) + + y) + »(a+/3) + (a/3 + Py+ay)(m+ x)} £ 

 — { A/3y + /J ay + j' a/3 + a |3y(ara+ =0. 

 The solution of this equation will be as follows : — 



where 



P=A + Ba? + Ca; 2 + ... + H*\ 

 Q=A'4-B'a?+CV+...H'^ 

 R=A" + B"^+CV+. . . HV. 

 where A, B, . . .A', B', . . . A", B" . . . are constants to be determined ; 



J_ 00 («-a) A+1 (^-/3)' t+1 (w— y) v+1 



Now let a be essentially negative, jS and y essentially positive. The^ 

 since the integral cannot always go on increasing with (#), we have Q=0, 

 11 = ; 



\v+l 



(w-a) x + , (w-/5)' t+1 (fc-y> 

 = Pe a *=(A + Ba7+(V+. . . +H^)e^': 



the constants A, B, C,. . where one of them is known, may be deter- 

 mined by substitution in the differential equation 



("i+*)^-{x+/*+*+tf+r-2«X«>*)}^ 



-{X(2a-/3-y) + /i(a~y)+Ka-/3) 



-(a 2 -a/3-ay + /3y)(m + ^)}^|-X(a 2 -a^-ay + /3y)y=0. 



But they are better determined in the following way. We may evidently, 

 without any loss of generality, put m = 0. In this case we shall have 



If we put #=0 in this integral, we have 



d . iz 



iy+\iz-py +1 (iz-yy +l ' 



