102 Mr murphy, ON THE RESOLUTION OF 



The equation <p{A) = A must here have two equal roots, suppose 

 each = c ; transform the proposed equation by putting u,, = v^ + c, 



.-. v,+ , = - c + (piv,, + c) = F{v,), 



the corresponding equation A = F{A) has then two roots each equal to 

 zero, and consequently F(A) must be of the form 



A -f mA' + nA' + kc, 



and accordingly F {v,,) or v^^^ is of the form », + mvf + nvj + &c. 



Hence if r„ vanish, «;,, v-,, V3, &c. successively vanish, and therefore 

 we put generally 



ill which series the coefficients are unknown functions of x, but 

 independent of «'„. 



Hence, «,.+ , = A, . i\ + B,.v^^ + C,.v^ + D^.v.^kc (1), 



and i', = F(i'„) = F, . i\. + F, . <-' + F,i\t + F, . v,' &c. 



where F„ is put for - — - — ^—^, by Maclaurin's Theorem. 

 '■ 1 . 2 . 3...ti 



VIZ. 



Heside the foregoing form of expressing r^+i, there are two others, 



F{v,) = t'„, = F, . r.. + F,.vJ + F-s . v.^ + F, . v/ &c (2), 



(V+l = A.r+, . V„ + J?.+, . V„' + C+l . i'o' + -Or+l ■ »o' &C (3), 



and since Fi is manifestly unity, if we compare tlie expressions 1 and 



2, when the latter is arranged according to tlie powers of r„, we obtain 



^, = 1, which is obvious by the law of the successive formation of the 



quantities t\, Vj., &c. ; also putting .r = in the general value of «,, 



we have v„ — v,. + -B„ . iv + C„ . v,? + D^v^ &c. , 



which shews that i?„ = 0, C = 0, &c. ; this being premised, we have 

 by comparing the expressions (1) and (3), the following identity ; 



V. + B,^,. v; + C+i . Vo + D.,, • i-o &c. = {vo + F,. v.; + F,v,' + F.v^' kc] 



