III. On certain formulæ for differentiation

In seeking for a formula in the theory of multiple definite inte­grals, I was several years ago led to investigate the successive differ­ential coefficients.of (√ x + λ ¯ ‒ √ x + μ ¯ ) 2 i , and the results which.I then obtained are given in my paper, “On certain formulæ for dif­ferentiations, with applications to the evaluation of definite integrals*.” I subsequently sought for the successive differential coefficients of the more general expression {( x + λ) ( x + μ) } ½ k (√ x + λ ¯ — √ x + μ ¯ ) 2 i , but the investigation was not finished. My attention was recalled to the subject by two remarkable identities obtained in Prof. Donkin’s memoir, “On the equation of Laplace’s Functions, &c.,”by a comparison of his results with those of Prof. Boole, which identities I perceived to belong to the class of formulæ above referred to : the first of the two identities is in fact readily deduced from a formula in my paper; the demonstration of the second is much more difficult, and I have only succeeded in making it depend on the establishment of the equality of the coefficients of two expressions of the same form. I have since resumed the unfinished investigation above referred to. The several results which I have obtained are given in the present memoir. I remark that, putting for shortness P=2 x + λ + μ, Q = √( x + λ) ( x + μ) ¯ , R=(√ x + λ ¯ ‒ x + μ) 2 ¯ , the subject to which the results all belong is the differentiation of the expression P α Q β R γ ; the before-mentioned expression {( x + λ) ( x + μ) ½ k (√ x + λ) ‒ (√ x + μ) ¯2 i is of this form, and the question in relation to it is to obtain the development of ∂ r x P α Q β R γ , where a = 0. The question arising from the second of Prof. Donkin’s identities is to obtain the development of (P ‒1 Q 4 ∂ x ) γ P α Q β R γ , where a —γ ‒ β. As the demonstration of these identities is one of the objects of the present memoir, I have given in the first section their reduction to the form in which they are considered. The second section treats of the development of the expression ∂ r x P α Q β R γ where a = 0; the third section of that of the expression {P -1 Q 4 ∂ x } r P α Q β R γ where a =γ—β; the fourth section contains the applica­tion of the formulæ to the demonstration of the two identities and some other applications of the formulææ.