208 
MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
and since it is visible that h may be expanded in the form A k + B k 2 +, &c., it will 
be unnecessary to consider in l any term after ^ 
d n u 
Hence is the coefficient of k n , when for h we substitute its value in terms of k in 
the polynomial. 
H = %Z.h’ + n 
d n ~ x u 
.h n -1) 
d n ~ 2 
. h n ~ 2 +...2.3.4 
du , 
n T,- h - 
h 3 
dx n • 1 '"dx *- 1 r v 
Again, & is given in terms of A by Taylor’s theorem. 
l _ £2 A + ^ + *2 -ff + &c 
Put for abridgement 
v dZ y (^iY l h » (Fy (^X~ l JYYlv (<hY , o 
1 ~dx 3 '\dx ) * 2 • \dx) ‘ 2.3*" dx 4 ' \dxJ ’ 2 . 3 . 4 0lC ' 
The equation for determining h in terms of k becomes then 
h ~ { k ( d rX l - h X =0 - 
In a memoir on the Resolution of Algebraic Equations, published in the fourth 
volume of the Transactions of the Cambridge Philosophical Society, I have proved 
the following rule. 
If <p (%) = 0 be an equation which contains only entire positive powers of x, and 
f (a?) any other function of the same kind, of which the differential coefficient or de- 
rived function is /' (x), then the value of f ( x ) will be found by taking the coefficient 
of — in the expression — t (x) 1. . — — . 
x L oc 
Applying this rule to the case before us we find that H is the coefficient of h m 
the formula 
-s-'j-frsr-v)}, 
and consequently is the coefficient of in the same formula. 
The first (n — 1 ) terms in the expansion of the logarithm do not contain k n and 
may therefore be neglected, instead then of 
we may use the series 
id • mr - + .-ms • m" -*r +,**.=*, 
and the value of 
d H . d U tri — 1 i / i \ 
is n ~rzji -h + n (n — 1 ) 
,re — 1 
d . u 
dh 
dx n 
dx 
ITT- » 
re— 2 
