61 



Hence taking the limits, as before, we have the differential equation. 



!f dx ^ \dx/ 



and its integral 



y =: nx + a^ becomes, because a = — ^ 



X 



Integrals of this kind are well known among mathematicians un- 

 der the name of particular integrals. 



Thus we have obtained from an equation to finite differences 

 two integrals each containing a constant arbitrary quantity, and 

 then by applying the method of limits we have from thence ob- 

 tained a differential equation and two integrals, one a conunon in- 

 tegral, and the other a particular one, and thus the method of 

 limits in this, as doubtless in all other cases when properly applied, 

 gives exact results. 



The elucidation of these and similar difficulties seems of some 

 importance, when it is considered that Lagrange, to whom un- 

 questionably belongs a very high place among mathematicians, per- 

 haps the highest, appears to have considered the method of limits with 

 less attention, than was due to it, and to have imagined it involved 

 in difficulties that do not belong to it. After having given the 

 preceding example he proceeds to remark on, and make similar 

 objections to, the method of limits, in other instances, and particu- 

 larly in deducing Taylor's theorem from the equation of finite 

 differences. These objections, as it appears to me, may be easily 

 obviated, and the only proof in every respect unexceptionable of 

 this important Theorem deduced from the limits of the equation of 

 finite differences. 



