140 Mr. Murphy on the 



Laplace's Theorem. 

 Let * = F\a + hf(i)\ to find /(si). 



Put z = F (a + u) the equation becomes u=k<p.F(a + u), to find 

 J'F(a + u), or f(z) apply the rule in Section (2). 



Hence 



f(z)=/F(a) - coefficient of - in f'F(a + u) 1. jjl-fl ^- F( " + ?<) ( 

 and expanding the log. as in the last case, it evidently gives 

 /(*) =fF(a) + kfF(a).<f>F(a) + £-{fF(a).jF~a)' + &c. 



Burmann's Theorem consists in this, that if as and w are two 

 functions of x which vanish together, and X any other function, 

 then 



d"X =dn . A'd^-\u)i 

 du" dz"- 1 



putting x and u = after the differentiations. 



In fact, if we form the equation « = AJT" +l , the value of u is 



i 



1 X" + '\ 

 coefficient of - in -1.(1 -h. ); 



therefore coefficient of A" +1 in u 



= coefficient of - in , =t — —\ = ==.rf«<" 



u (n+l).M" + 1 1.2...W + 1 



when u is put = by Maclaurin. 



Put the same equation under the form 



z = h . * . X^\ 

 then by Section (2) 



" 



« or hX n+ '= coefficient of = in - h^— 1.(1 - - . - X" +1 ); 



z dz x z u ' 



