A new Demonstration of Taylor's Theorem. 229 



I annex was given me at one of my lectures in our University by 

 an undergraduate, whose knowledge of the calculus was limited 

 to the first twenty sections of Lacroix. It is independent of the 

 theorem of Maclaurin, and gives that series as a, particular case. 

 In this respect, amongst others, it excels the proofs of Lacroix. 

 I am Sir, &c. 



Dionysius Lardneu, A.M. 

 University of Dublin. 



Demonstration of Taylors Theorem, by Edward Wilmot, Esq. 

 Trinity College, Dublin. 



Let n = F (x) and n — F (x + h) ; let 



»'=A + B(x+A) + C (a+A)»+D [x + h) 3 + E [x+hy 



Where A, B, C, . . . . are independent of x and h. The powers 

 of (x + h) being expanded, and the series disposed by the dimen- 

 sions of h, it becomes 



n = (A + B x + C& . . . ) + (B + 2Cx + 3Dx 9 . . ) 



h + (C + 3Dx + 6Ex 9 . . ) h* 



The successive co-efficients of this series are evidently 

 dn d*n 1 d 3 n 1 d*n _1 



"'to' -d*\ T2 ' dtf ' 1.2.3' dx* 1.2.3.4 



:nce we have 



, dn h , d*n h* , d 3 n _A 3 _ 



n ^ n + dx-T + ~d^ , T2^ d7; i.3.3 T 



By supposing x = o in this series, we find that of Maclaurin. 



Art. V. Historical Statement respecting the Liquefaction of 

 Gases. By Mr. M. Faraday, Cor. Mem. R. Acad. Paris, 

 Chem. Assist, in the Royal Institution, &c. 

 I was not aware at the time when I first observed the liquefac- 

 tion of chlorine gas*, nor until very lately, that any of the class of 

 bodies called gates, had been reduced into the fluid form ; but, 

 having during the last few weeks sought for instances where such 

 results might have been afforded without the knowledge of the 

 * Phil. Transactions, 18'J3, pp. 1G0.I89. 



