94 



grals*." I subsequently sought for the successive differential 

 coefficients of the more general expression { (x + X) (x +/*)}**( */ x + \ 

 */ x + /u) 2 , 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.f," by a comparison of his results with those of Prof. Boole, which 

 identities I perceived to belong to the class of formulae above re- 

 ferred 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 



, R=( ^# + A- ^Tj") 2 > the 



subject to which the results all belong is the differentiation of the 

 expression P a Q^ R v ; the before-mentioned expression {(#-f\) 

 (#-f /u)* A ( 4/a?-|-\ V#+ju) 2t is of this form, and the question in 

 relation to it is to obtain the development of p* P a Q^ R y , where 

 a=0. The question arising from the second of Prof. Donkin's 

 identities is to obtain the development of (P" 1 Q 4 d.,.) Y P a Q^R v , 

 where a=y /3. 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 dP a Q^ R v 

 where a=0 ; the third section of that of the expression {P" 1 Q 4 b x } r 

 P*Q^R V where a=y /3; the fourth section contains the applica- 

 tion of the formulae to the demonstration of the two identities and 

 some other applications of the formulae. 



* Cambridge and Dublin Mathematical Journal, t. ii. pp. 122, 128 (1847). 

 f Philosophical Transactions, 1856, pp. 43-57. 



