EXPANSION OF F {x + K). 157 



which is derived from (1) by a simple transformation, R taking the place of 

 h. ^' {x . h), and consequently vanishing with h, is sufficient to establish all 

 the rules of differentiation. 



Observing that in the preceding investigation x was confined within certain 

 limits, while h remained arbitrary, we may replace re by a number r within 

 the limits supposed, and hhj x — r which denotes the variable difference be- 

 tween the general and special values of a;. Equation (2) will then become 



u = Fx = Fr +p [x — r) +p' {x — r)" + p" {x — rY + M; 



where p p' p", &c., denote the values of P P' P", &c., when x = r, and 3/ the 

 value of If. " Q when x = r, and h = x — r. 



Differentiating successively, and denoting the differential coefficients of M 

 by M' M", &c., we have 



^ =p + ap' {x-'r)''-' + bp" {x — r)'-'+ M, 



Ui X 



^=a{a—l){x — r)''-' + bib—l)p"{x — r)''-'+ M", (3) 



CI ►o 



^^ = a{a—l){a — 2){x — r)''-' + b{b — l){b — 2){x — ry-' + M"'. 



We may suppose that each term in these equations is the only one which con- 

 tains the power of x — r peculiar to it, for if there were several terms contain- 

 ing the same power of x — r, they might be united into one. 



Observing, now, that the exponents a, b, c, &c., are each greater than unity, 



and are arrang^ed in ascending order, if we make x = r, and suppose -,— , -^ — -, 

 ^ o ' ' ff dx dx^ 



&c., to remain finite, the first of equations (3) becomes 



'dii^ 



fau\ 

 \dxj 



P\ 



a result already established, and implied in the process of differentiating. 

 With regard to the 2d equation we must have 



a > 2, or a < 2, or a = 2. 

 If a > 2, every term in the 2d member antecedent to M" will vanish, and if 



72 



M" does not vanish, it must either be finite or infinite. But, since (-n — ) is 



^dx'^ 



finite, M" cannot be infinite. If it reduces to a finite quantity A, then M"^ 



* See Mr. Bonnycastle's paper, pages 245, 246, Vol. VII. of these Transactions. 

 VIII. — 2 p 



