284 Mr. W. H. L. Russell on Linear Differential Equations. [Feb. 2, 



y essentially negative. Then the integral will satisfy the differential 

 equation 



_ { X( / 3+r)-^(a + y)-K^+«-(a/3 + /3 r -f ar >}g 

 — {\(3y + juay -f vafi + oc(3yx}y — - 0. 

 We here suppose X fractional, and v positive and entire. The solution 

 of the differential equation will be 



2 / = PeP* + Q e y*+C{Pe^X 1 € 2 ^+QevyX 2 e 2 ; 



where P and Q are rational and entire functions of the orders /x and v re- 

 spectively, and X 1} X 2 functions of x, which it will be needless to write 

 down. The integral will consequently be equal to this expression when the 

 arbitrary constants are properly determined. The integral cannot go on 

 increasing as a?, supposed negative, increases. Hence Q=0 and C— 0, and 

 the integral becomes 



^ frt-Z-yytf =(A+B*+(V + . . . + 



where A, B, &c. are constants to be determined. 

 We find, as before, 



I 



A - r°° (*-«)*-*& 



(z— a.y- x dz 



z-py(z- y y+i~' 



&c. == &c. 



And thus the integral is completely known. 

 I will now consider the differential equation 



dx dx z dx 

 It is easily seen that this equation is satisfied by the integral 



e- nu2+ ™du 



y= L • 



The solution of the differential equation is 



y=C 1 cos ax + C 2 sin ax+ C 3 {cos axfe* 2 sin ax dx— sin axfe* 2 cos ax dx) ; 



and these must be equivalent. 



If we put — u for u in this equation, we easily see that 



1 a* + w 2 1 « 2 + w 2 i 



J -CO 00 



hence the integral is unchanged if — # is put for and therefore C 2 = C 3 = 0. 



