l6 REPORT 1872. 



form +2/, it gives no indication of its value when n is not a positive or negative 

 odd integer ; and it will )je found that the two most natural methods of evaluating 



this integral, viz. by expanding the factor e ''' and integrating term by term, or by 

 multiplying by e^" and transforming the new integral, &c., both fail through the 

 occurrence of infinite values for the terms after a certain point, the reason for which 

 will appear further on. It might perhaps be thought that when n was arbitrary 

 the formula (1), the factorial being replaced by the Gamma-function, would still 

 be true by the principle of the permanence of equivalent forms, the series then 

 extending to intiuity ; but such is not the case. The value of the integral in the 

 general case may be found as follows, the steps of the method only being indicated 

 in this abstract." 



It is found that Eiccati's equation ^-x-^~^uz=0 is satisfied by the integi'al 



e~"dz, u being written for z ^+ ~rT*' ^^^ ^^®° by certain series given in the 



' Philosophical Magazine,' ser. 4, vol. xxxvi. p. 348. As the differential equation is 

 linear, it follows that the integral must be of the form A x one series + B X the 

 other series, A and B being constants. Transforming every thing now by assuming 



1 «' 

 «=-, a = — , it will be found that, after very considerable reductions, we have 

 q 2q > 



the result that if 



U=l- 



(2a-y 



-2 {n-2)(7i-4:).1.2 

 and 



«t2 (n + 2)(«+4j.l.2 '■■' 

 then 



j v" e "' ^•Vr=AU+Bn'"V (2) 



[The details of the transformation indicated abo^e are, to a great extent, given in 

 the ' Pliilosophical Magazine,' for June 1872 (vol. xliii. p. 433 &c.). The original 



q 



series are given at the top of page 434, and their transformations (taking — = /3) 



at the bottom of the same page, while U and V are merely (:?) and (3) of page 

 435, 2a being written for /3, as is done throughout. The foilowing errata should 

 be noticed in the formulre as they stand in the ' Thilosophical Magazine,' viz. the 

 factor /3" is accidentally omitted from the values of 11 and S given at the foot of 

 page 434, and the factor 2 is omitted in the denominators of the second term in 



(2) and (3) ^it should be ^T—p^ ) ; also in (2) /3^ should be fi\ None of these 



slips affect the subsequent work, for they are treated as if in their correct forms, 

 not as printed.] 



Eesuming (2), it remains to determine A and B. By putting a=:0, we obtain 



at once A = ^r r^Y Let B=^(«); aiid transform (2) by talcing " for v ; we thus 

 find ~ " 



J" 



-n + l -v'i- 



V e 



^d« = ir(«)a-"U + (^(«)V (3) 



But this integral is the same as the integral in (2), with the sign of « changed ; 

 Therefore, observing that a change of sign in n turns U into V, and vice versd, we 

 see tliat the right-hand side of (3) also 



* Quarterly Journal of Pure and Applied Mathematics, vol. si. p. 267. 



