60 The Rev. C. Graves on the Calculus of Operaiions. 



at which Professor Challis has arrived with reference to ray 

 vibrations. I have done so partly because the subject has been 

 taken up by the Astronomer Royal, partly because I have not 

 leisure for the discussion at present, partly because the points 

 which would have to be noticed, and which would be likely 

 to arise in the course of the discussion, are so numerous, that 

 I think it hardly fair to take up the pages of a Magazine like 

 the present with the controversy. I cannot however conclude 

 without recording my protest, first against equation (8.) 

 (vol. xxxii. p. 282), and secondly against equations (B.) and 

 (C.) (vol. xxxiii. pp. 99 and 100), by which an attempt is made 

 to satisfy equation (A.). 



Pembroke College, Cambridge, 

 Dec. 23rd, 1848. 



VII. On the Calculus of Operations. By the Rev. Charles 

 Graves, M.A., Professor of Mathematics in Trinity College^ 

 Dublin *. 



PROFESSOR YOUNG, objecting to the method by which 

 the theorem of Leibnitz is usually extended to successive 

 integration, has lately proved its applicability in that case by 

 means of repeated " integrations by parts : " and he has shown 

 how to obtain in this way a series of supplementary integrals, 

 without the addition of which the theorem is not generally 

 true, though they are commonly suppressed in the statement 

 of it. Professor Young seems to impute this omission to the 

 nature of the Calculus of Operations, by means of which the 

 theorem is usually treated ; as though that method necessarily 

 gave the theorem in the imperfect form, and made no provision 

 for the correction which he suggests. 



It is my purpose here to show that the omission of the sup- 

 plementary integrals has been caused by the use of an incom- 

 plete form of the binomial theorem, rather than by any inhe- 

 rent deficiency in the Calculus of Operations, which, if applied 

 to this problem with proper caution, will furnish the desired 

 result in a direct and elegant manner. 



If we take the identity 



{l+x)-'^=:l—x + x'^ — 8ic +(-!)'»- 1^;'"-' 



+ ( — 1 )'"(!+ J^)"'^"*, 



and differentiate it (n—l) times; we shall obtain a develop- 

 ment, which coincides for its first m terms with that obtained 

 by the use of the binomial theorem ; and furnishes moreover 



* Communicated by the Author, 



