REPORT ON CERTAIN BRANCHES OF ANALYSIS. 215 



polations ; and subsequently, by a much more direct process, 

 which lead to the equation, 



r (r) r (1 r) = -r^ — (when r > < 1) : 

 ^ < ^ sin r ir ^ 



If, under the same circumstances, r be greater than n, the coefficient of dif- 

 ferentiation becomes infinite, and its value, determined as above, becomes 

 x'—» ,, , Ci a*"-"-' 



= roOVir-n + l) {'°S*' + ^} + r (r - u) + ^'=- 



^iWrcT^TTTT) ^^"° " + ^^ '^' "" ^'•-«+i) r («-r) + c} 



(r — n) 

 which is in a form adapted to all values of r. 



The cases which we have considered above are the only ones in which the 

 coefficient of differentiation will become ivfinite, in consequence of the intro- 

 duction of log « in the expression of its value. We shall have occasion here- 

 after to notice more particularly the meaning of infinite values of coefficients 

 as indications of a change in the constitution of the function into which they 

 are multiplied. 



(6.) Uti = (a X -I- i)«, then 



dru r(l-J-w)«'' . .V Ci"--! 



mrr = Y\\--\-i^-V) • («* + ^y-" + rT=7) + • • • • 



For if V = a X -j- b, then -j— = a and -7— j = ; and therefore 

 d^ u T (\ ->t- 7i) /dv\- , C«'-> , „ 



Thus if M = (a; -|- \y, we get 



rf«^ ^ ^^' V (- i) x" r (- 4) a;^ 



8 .„, ,^4 , C , C, , C2 



(x+ir + ^ + ^ + ^ -I-&C. 



*^ X- x' X- 



If we replace (a; -f 1)2 by a'2 -)- 2 -|-a; 1, we shall get 



^ _ L(^) ^.^ , 2 £11) J 4. Eil) _L 

 d„4 - r a) '^ + 2 J, ^^^ * + r (i) • x4 



C Ci 



r (— \) x^ r (— 4-) «* 



It thus appears that the two results may be made to coincide with each 



other, when (ar -)- 1)t jn the fi.rst of them is developed, by the aid of the proper 

 arbitrary functions. 



The necessity of this introduction of arbitrary functions to restore the re- 

 quired identity of the expressions deduced for the same differential coefficients, 

 presents itself also in the ordinary processes of the integral calculus : thus, 

 if u = (a; -I- 1)2, we find 



