74 ROYAL SOCIETY OF CANADA 



The cases where a = 0, a := 1, and a = 2, are of particular importance 

 in the class of problems already referred to. Introducing these values 

 Equation (6) becomes^ 



a = 0. Ku = —fe-'dx — ^fe'"" x^ dx (7) 



a = \, Ku= ~ f- ■' 'i-^ — ^A""''-^ '■^^ + -?/«"'' ^'^^ (^) 



J X ? «^ 'i «^ 



« = 2, 7?. ^ - /^f ^.r - ~Je-'- dx + ^/6-'- ^ dx 



_ .l+ilVe--^- x2 <^.r (9) 



and by means of tables of these integrals computations can be carried out 

 with much less labour than if the method of integration by partial fractions 

 — assuming its applicability — were employed. This is particularly the 

 case when the factors in Equation (1) are numerous, or the exponents 

 large. 



The most general form of Equation (1) may be written 



Kdu = 



—dy 



y'' (1 + M + Kf + ^y + . . 0^^ (1 + ^22/ + hy' + m,t/ + . . .)y . . 

 rom which the following values of g', r, s . . . .may be found 



q = 2/3g ; r = h ^ftcf — 2/iA ; s = ^ ^7%' — ^1^9^ + ^Y^^i 



A^ f:i, y . . . may be negative, this form includes the cases where 



factors are contained in the numerator as well as in the denominator. 



When the indices are fractional, the method here described aftords a 



means of integration where others fail ; it is, however, as already stated, 



1 If e^ be substituted for e'^, the sign of the variable limit must be reversed, and 

 that of the integral may be affected. Thus : 



-X X 



l^dx = f^I^dx 



J x i X 



-X X 



-00 CO 



f e^ dx = — / c"* dx 



-X X 



-» 00 



I' e^ xdx = J e-^x dx 



-X X 



fe^x'dx = — f e-^x'^dx 



