CHAPTER II. SUCCESSIVE DIFFERENTIATION. cases. The analogy between Algebraic powers and successive differentials, when expressed by the notation of Leibnitz, was observed soon after the invention of the Calculus. Leibnitz himself paid much attention to this subject, as may be seen in his correspondence with John Bernoulli; and, in the course of his investigations, he discovered, by induction, the Theorem which bears his name. He also conceived the existence of differentials with fractional or irrational indices, but he made no steps towards the calculation of such functions in any In recent years that branch of the Calculus has acquired considerable importance, and it appears to be the quarter from which we may look for great additions to our knowledge of analysis. I shall however in this chapter confine myself to examples of differentiation with integer indices, partly because there are still some points in the theory of general differentiation which are not entirely fixed, so that the subject is not adapted for the student; partly because the principles of that branch of the Calculus are not laid down in any Elementary Treatises which a student could consult, and it would occupy too much space to enter at large on the subject in the following pages. Those who wish to see the results of the labours of mathematicians in this field of research are referrred to various Memoirs of Liouville in the Journal de l'Ecole Polytechnique, Vol. x111., and in Crelle's Journal ; to two papers by Professor Kelland in the Transactions of the Royal Society of Edinburgh, Vol. XIV.; to Professor Peacock's Report on the Progress of Analysis in the Transactions of the British Association; and to two papers by Mr Greatheed in the Cambridge Mathematical Journal, Vol. 1. du * cosø {cos (x sin ) cos 6 - sin (v sin ) sin e} da Murphy, Cambridge Transactions, Vol. v. p. 342. In functions consisting of the product of two or more simple functions, we may make use of the Theorem of Leibnitz, the enunciation of which is as follows. Commer. Epis. Leib. et Bern. Vol. 1. p. 46, 99. d' (uv) = n(n − 1)... (n-r+1)(1 – X)"z"-{1 dx" n - 1+11 - V d' (uv) r(n-1) =put {a”x+r.na"-120-1+ n(n-1)a"-2,4-+&c.}. dw" 1.2 In the same way, if uv = cat ai", duo) 1 n - po + 1 r(-1) 1 + po (r – 1) (r – 2) 1. + &c.} 1.2.3 (n - p + 1)...(n - q + 3) If p = n, n d" (20" logx) =n(n − 1) 3.2.1 {log x + dx" 12 n (n − 1) (n − 2).1.2 (1.2.3) + U V = n (a + x)" (16) (c + a)" d' (uv) (a + x)*-* a + x = m(m – 1)...(m-p+1) {1-1 da" (c + x)" 1 m - 9+1c + r(n-1) n(n+1) (a + x) + &c.}: 1.2 (m-r+1)(m-r+2) (c + x)? |