210 THIRD REPORT — 1833. 



■it to be permanent, so long as we do not at the same time as- 

 sume the necessary existence and interpretation of equivalent 

 results. If, however, such results can be found, either gene- 

 rally, or for particular values (not integral and positive) of r, 

 apart from the sign of integration, the consideration of the values 

 of the corresponding differential coefficients will involve no 

 other theoretical difficulty than that which attends the transition 

 from integral to fractional and other values of common indices. 



Euler, in his Differential Calculus *, has given the name of 

 inexplicable functions to those functions which are apparently 

 resti'icted by their form to integral and positive values of one or 

 more of the general symbols which they involve : of this kind 

 are the functions 



Ix2x3x X, 



I '^ 2 '^ 3 '^ x' 



111 1 



1 H i. — 



1" 2" ' -^" 



1 4- " ~ ^ I ^ ~ ^^ a — (x — 1) b 



'^ a + b '^ a + 3b '^ a + {x + l)b' 



and innumerable others which present themselves in the theory 

 of series. The attempts which he has made to interpolate the 

 series of which such functions form the general terms, are 

 properly founded upon the hypothesis of the existence of per- 

 manent equivalent forms, though it may not be possible to ex- 

 hibit the explicit forms themselves by means of the existing 

 signs and symbols of algebra. In the cases which we have 

 hitherto considered, the forms which were assumed to be per- 

 manent had a real previous existence, which necessarily re- 

 sulted from operations which were capable of being defined. 

 In the case of inexplicable functions, the corresponding perma- 

 nent forms which hypothetically include them, may be consi- 

 dered as having an hypothetical existence only, whose form 

 degenerates into that of the inexplicable function in the case of 

 integral and positive values of the independent variable or va- 

 riables. It is for the expression of such cases that definite 

 integrals find their most indispensable usage. 



* Instiiutiones Calculi Differentialis, Capp. xvi. et xvii. See also an admira- 

 ble posthumous memoir of the same author amongst the additions to the 

 Edition of that work printed at Pavia in 1787- He had been preceded in such 

 researches by Stirling, an author of great genius and originality, whose la- 

 bours upon the interpolation of series and other subjects have not received 

 the attention to which they are justly entitled. 



