434 Mr. Murphy on the General Properties 



in the ordinary method by putting e""" for /(a), the equation for 



determining the values of m is €'"''' — 1=0; now this reducing 



equation which is obtained only from the left side of the linear* 



equation, can never be the same with respect to two linear 



equations, unless their left sides be identical. But in this case, 



^, , df{a) .,. , 



smce (f> (a) = -^ — , it we put 



rt« 1 an 3 da 



f„ia) being used in a similar sense to (p„{a), the resulting equation 

 on the supposition of U=u, will be exactly identical with e""*- i =o, 

 as is obvious from the kuown equation 



2 mh 



= 1 4 



1 —mh 



1 + ^ mh 



1 —^mh 



1 +&C. 



Theorem IV. If a, a + h, be the limiting values of x, and « any 

 quantity whatever, then shall 



D.<f){x)=h\<p{a+a)+ -Y^.<p'{a + 2a)+ -^-^ . cp" {a + 3 a) 



0'(a), (p" (a), &c. being the differential coefficients of 0(ff). 

 For the terms in the preceding series which contain 



«/)'" + "'' (a), o", 

 when respectively expanded by Taylor's theorem, are those after 

 the m* place, and if the coefficient of that quantity be collected 

 out of these different expansions, its value is 



1.2...WI 



_____ S^{m + 1)-' -« (m + 2)— + ^:^ (m + 3)"-', &c.l , 



