32 On the Property of the Parabola 



But by Taylor's theorem the same quantity is tlie coefficient 



h* 

 of ,-A a > in the expansion for U — w, we therefore find 



the theorem announced at the commencement, viz. 



da^ L 1*2.3 da] 



We have not space here to point out the applications of 

 this general theorem, and shall therefore close this paper with 

 two remarks. 



First, if (p [x, y) be of the form x (p (y\ then A <|> = ^ 45 y ; 

 we have then 



?,{(..f.^:}acc.; 



h^ ^ 



1.2.3 d. 



and if we suppose jc = 0, then y = a^ and U is then the 

 value ofy(y) determined from the equation yz= a + h<p {y): 

 we thus fall on Lagrange's theorem. 



Secondly, that if the proposed equation were 



^ = F {« + <p (JT, 2/)} and M =/(j/), 



the fundamental equation (1.) would remain the same, and 

 therefore this theorem admits of the same extension that 

 Laplace gave to Lagrange's. R. M. 



VI n. On the Property of the Parabola demonstrated by Mr. 

 Lubbock in the Phil. Mag, for August, By A Corre- 

 spondent. 



To the Editors of the Philosophical Magazine and Journal. 



Gentlemen, 



1 N the last [August] Number of your Journal there is a de- 

 ■*- monstration by Mr. Lubbock of a very beautiful property 

 of the parabola. Mr. L. does not seem to be aware that this 

 problem, which he ascribes to the French, is in fact due to 

 Prof. Wallace of Edinburgh, who before the end of the last 

 century communicated it to Mr. Leybourne, by whom it was 

 published as a prize question. Not having the book at pre- 

 sent, I cannot tell in which volume of the Repository it is to 



