250 ON THE INSUFFICIENCY OF TAYLOR's THEOREM, ETC. 



others, I may mention one readily deduced from the above which that excel- 

 lent analyst Mr. Murphy has given in the Philosophical Transactions. 

 The theorem above given may be conveniently put under the form 



h "^ -ne^^ 



^ r ^ __ F (Y + ic9v/~i)d^, 



h -«v'=-i 



0|27r 



where the limitation is that — shall be less than unity, and h and k each suffi- 



k 



ciently small to make F (x -f h) and F (x + k) converge. 



NOTE. 



While this paper was going through the press, the ingenious and distinguished author fell a 

 victim to a disease under which he had long suffered; his death occurring on the 31st of October, 

 1840. The first proofs only were subjected to his examination, and he was then so ill that some 

 apology may be necessary for errors of the press in the earlier, as well as the later parts of the 

 paper. Mathematicians will be struck by the circumstance that the formulae are printed in Roman 

 instead of Italic letters, — a deviation from the usage which was not observed by the author until it 

 was too late to have it corrected. 



