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.
III. On certain formulæ for differentiation
