208 THIRD REPORT— 1833. 



using the simple symbol r both in one case and the other. In 

 a similar manner, in the expression of Demoivre's theorem 



(cos 9 + \/^^l sin fl)" 



= cos (2 ?• w TT + ?i fl) + -v/ — 1 sin {2 r mr + n 6), 



we may suppose w to be any quantity whatsoever *, but r is ne- 

 cessarily a whole number. 



In some cases, however, the construction of the formula it- 

 self will sufficiently express the necessary restriction of the 

 values of one or more of its symbols, without the necessity of 

 resorting to any convention connected with their introduction : 



thvis, the formula 1 x 2 x 3 r, commencing from 1 , is 



essentially arithmetical, and limited by its form ro whole and 

 positive values of r. The same is the case with the formula 

 r (r — 1) . . . . 3 . 2 . 1, where some of the successive and strictly 

 arithmetical values of the terms of the series r, r — 1, &c., are 

 put down ; but the formula r (r — 1) ('' — 2) .... is subject to 

 no such restriction, in as much as any number of such factors 

 may be formed and multiplied together, whatever be the value 

 of r. In a similar manner, the formula 



n {n — I) ... {n — r + I), 

 r72 77. r 



which is so extensively used in analysis, is unlimited with re- 

 spect to the symbol w, and essentially limited with respect to 

 the symbol r : it is under such circumstances that it presents 

 itself in the development of (1 + a;)". 



In the differential calculus we readily find 



'^ = n{n-l). ..{n-r + l)x"-', 



and in a similar manner also 



•^(.r« + C, x'^-' + C, x'-^ + . . C„) = w (w-1) . . (H-r+ 1>«-'-: 

 dx^ 1 



in both these cases the value of n is unlimited, whilst the value 

 of r is essentially a positive whole number ; in other words, 



* The investigation of this formula (like the equivalent series for (1 -|- x) 

 when n is a general symbol,) requires the aid of the principle of the perma- 

 nence of equivalent forms, in common with all other theorems connected with 

 the general theory of indices. The formula above given involves also impli- 

 citly any sign of affection which the general value of n may introduce : for 



(cos 6 + V"^ sin tf)" = (1)" (cos n6+ V^-i sin n 6) 



= (cos 2 /• M TT + -\/— 1 sin 2rn-!r) (cos n6 + -v/ — 1 sin n 6) 



= cos {2 rn TT -{■ n d) -\- V' — 1 sin (2 r n rr -\- n 6) 



