KEPORT ON CERTAIN BRANCHES OF ANALYSIS. 211 



It is easy to construct formulae which may exhibit the possi- 

 bility of their thus degenerating into others of a much more 

 simple form, when one or more of the independent variables be- 

 come whole numbers : of this kind is the formula 



« + /5 sin (2 rTT + 5) + y sin (2 r ff + 6') + &c. ^ , . 

 « + /3sin9 + ysinfl' +,&c. ^^ ^' 



which is, or is not, identical with <p (r), according as r is a whole 

 or a fractional number : such functions are termed undtdating 

 functions by Legendre *. We can conceive also the possible 

 existence of many other transcendents amongst the unknown 

 and undiscovered results of algebra, which may possess a simi- 

 lar property. 



The transcendent 



X^^O'^D" 



mentioned above, possesses many properties which give it an 

 uncommon importance in analysis, and most of all from its fur- 

 nishing the connecting link in the transition from integral 

 and positive to general indices of differentiation in algebraical 

 functions. If we designate, as Legendre has done, 



J^' dx(log^yhyr(l+r), 



we shall readily derive the fundamental equation 



r(l + r) = rr{r)j- (1) 



which is in a form which admits of all values of r. It appears 



* Traits des Fonctiom Elliptiques, torn. xi. p. 476. 

 f In as much as 



_ r(l + n) , 



A x'-. 



and 



d"-r + » r (1 + n) 



r (1 + r) "- 



a:»-J = B .«»■-» = r A «»-», 



it follows that r A = B, and therefore also that 



which is the equation (l) : and it is obvious that the transition from 



a x^-^ dx^~^ + 1 



(which is equivalent to the simple diiFerentiation of A x^, when A is a 

 constant coefficient), will lead to the same relation between F (1 + r) and 

 r (r), whatever be the value of r, whether positive or negative, whole or frac- 

 tional. Legendre has apparently limited this equation to positive values of r, 



