= Ca; + C 



REPORT ON CERTAIN BRANCHES OF ANALYSIS. S13 



between any two successive whole numbers, they can be der 

 termined for all other values of r. Euler * first assigned the 



number, it is always zero : if r = — 1, it is finite when m = 1 : if r := — 2, 

 it is finite when >j = 1 or « = 2 : if r = — 3, it is finite when « = 1, or w = 2, 

 or M = 3 : and generally if r be any negative whole number, there will be 

 finite values corresponding to every value of n from 1 to — r ; wo thus get 



rf-'O _ n 



dx-i 



dx-^ 



3 5 = r-S + Ci a; + C2 



d x-^ 1 . 2 



d-« _ C A-"-i , Ci . ^"-2 



d^»- 1.2. .(».-]) + 1.2..(«-2) +U-2.* + U-i. 

 This is the true theory of the introduction of complementary arbitrary func- 

 tions in the ordinary processes of integration. 

 More generally, if r be not a whole number, 



d^O _ _C_ Y{\-n ) ^_„_^ 



dx''- r (0) ■ r(i — M — ?•) 



which will be finite when n is a positive whole number and when 1 — m — r is 

 not a negative whole number : thus if w be any number in the series 1, 2, 3 , . ., 

 and iir=k, then 



d x^ x^ x^ a; - 



In a similar manner, we shall find 



.fi r\ 3 1 . C2 . C3 .... 



? 2 = C a;^ + Ci a^ H r "I T + &c. tw tvjimtum, 



a a; - 

 The knowledge of these complementary arbitrary functions will be found of 

 great importance for the purpose of explaining some results of the general 

 differentiation of the same function under different forms which would other- 

 wise be irreconcileable with each other. 



(3.) The differential coefficient will be zero, when n is 7iot, and whenn — r 

 is, a negative whole number. 



Thus, 



d^x „ d3«2_ d^ x^ _ dJ x~^ _^ d~^ ^~ ^-(\. 

 = 0, — — 0, — 0, u, — _ u . 



d x^ dx^ 



d x^ d x^ dx 



_ 5 



7 



and similarly in other cases. 



(4.) The differentials of 00 are not necessarily equal to », but may he finite. 

 If we represent 00 by C F (0), we shall find 



Commetitarii Petrop., vol. v. 1731. 



