94 Mr murphy, ON THE RESOLUTION OF 



Now similar difficulties opposing the expansion of other kinds of 

 successive functions, these can only be removed by attending carefully 

 to the equations in finite differences, by which the law of the formation 

 of the functions is expressed. I therefore here propose a means for 

 resolving such equations, of whatever order or degree, in an algebraical 

 form. 



FIRST CLASS OF EQUATIONS. 



All successive functions such as those above mentioned, are repre- 

 sented by the equation «,„.,=^ («,,), for this manifestly expresses the law 

 of the successive functions, 



Mo = «, u,=(p{u), u, = (p(p(a), ih = <li(p<p(a), 



itj. = (p(j)(p(p (.1- times) {«}; 



hence the expansion of ti, in every such case depends on the solution 

 of this equation. 



Put u, =f{y') the form of the function /, and the quantity 7 re- 

 maining at present unknown, and also let 7^ = ss. 



Then, since ti^ =/{«), and u.,+ , =f{yz), we have 



Suppose wow f{z) = A + B% + Cz^ + Dz^ + &c. 



the preceding equation becomes 



A + Byx + Cy^%° + Dy^%\ kc. = <p {A + Bs + Cz' + Dz% &c.} 



and it remains to expand the latter function, in order to compare like 

 powers of z, and thus obtain the assumed coefficients and the quantity 7. 



Now by Taylor's Theorem the development will be of the following 

 nature, viz. 



(p(A) + Z,<p'{A) + Z,(p"{A) + Z,^"'(A) + &c, where Z,, Z„ Z„ &c. 



are functions of s completely independent of the form of the function (j>, 

 and <p'{A), (p"(A), &c. represent the successive differential coefficients 



of (p(A). 



