REPORT ON CERTAIN BRANCHES OF ANALYSIS. 249 



Thus, if in the triangle A C B, we 

 draw C D, a perpendicular from the '\C''* 



vertex to the base, and if we suppose X^. 



A D = a, A B = J, A the origin of 

 the coordinates, A B the axis of a", 

 y ■=. a. X the equation of the line A C, 



and y = a' ^ + /3' the equation of the ■* 



line B C, then we should find that the value of y represented 



by the equation 



y — ""Da . « ^ + ^Dj" (a' X + ^')* (2.) 



would be confined to the two sides A C and B C of the triangle 

 ABC, excepting only the point C, which corresponds to the 

 common limit of the discontinuous signs. For if we suppose 

 'Do" and -^Dj" to be true up to their limits, we shall find, when 

 X = a, that ^Do" + ■^D/ = 2. If we replace, however, 



'D; by ^D^" - ^, and -D," by 'D." - ^, 



Developement of the Binomial Theorem, which was published in 1827, have 

 contended vigorously for the restriction of the meaning of the sign =: to simple 

 arithmetical equality, and would reject its use when placed between a function 

 and its developement, unless its complete remainder, after a finite number of 

 terms, should replace the remaining terms of the series ; or unless, when the 

 indefinite series was supposed to be retained, the value or the generating func- 

 tion of this remainder could be assigned. In conformity with this principle 

 they have assigned the remainder in the series for (a + x-)", which they exhibit 

 under the following form : 



(o + •»)"=«» + M a"- >« + « (» - 1) • • • (w — r + 1) . ^„_,. ^ 



1.2... r 



{r + 1) a 



+ ^'"'('' + *)"{(^+V^ + ^ 



(a + xy + ^ 



Cr 4- I ■> Cf 4- 9'> r«_^^ «n- 



(r + 1) (r + 2) . . . (n - 1) a'-'-i ^ . 

 1 . 2 . . . (w — r — 1) ■(« + .«)»/' 



the remainder being (a + x)» x'' + 1 multiplied into n — r terms of the devc- 



1 1 



lopement of 77 — ; — r TTTT, or of — -rr- 



The method which they have employed for this purpose, which is extremely 

 ingenious, succeeds for integral values of n, whether positive or negative, but 

 fails to assign the law when the index is fractional. But my own views of the 

 principles of symbolical algebra would, of course, induce me to attach very little 

 value to results which were exhibited in such a form as to be incapable of being 

 generalized, a defect under which the formula given above evidently labours. 



• The conventional sign "D^ might be replaced, though not with perfect 



propriety, by the definite integral „_f, I dx. 



