RESOLUTION OF ALGEBRATCAL EQUATIONS. 29 
highest order in equation (2), be, U, V, ...... , Y; and let the sum 
of A and of those terms, such as ET, in (2), which do not involve 
surds of the highest order present in (2), be H. Then 
Ee ry BU CV sin. .d2..'..2 DY S/O. 
Again, let 
U = Fy Xe, Vv = H, INE ir hie aht ior} Y = H, iXiin'- 
where X, is the continued product of those factors of U, which are 
powers of surds of the highest order in (2) ; X2, the continued pro- 
duct of those factors of V, which are powers of surds of the highest 
order in (2); and soon. Then, putting 
eB Xi CH, Molt a... LEK On ak (4) 
let us suppose, if possible, that no such equation as (8) can subsist ; 
and, in connection with this supposition, let us make the hypothesis, 
that the terms, X,, Xs, &c., are all distinct from one another. By 
differentiating (4) with regard to p, we get 
d{ log (H) } d } log (BH, X,) } 
ae gt 
Multiply (4) by the coefficient of BH, X, in (5), and subtract the 
product from (5). Then 
hH +h, CH, X, + .....0. zea Up $ bY. Ge 0) per pee naa ae a (5°?) 
where the values of hf, h,, &c., are 
Ge) == Ores 5.0. (3) 
Et hae: Gil 
{ es (oi: x.) 
Pa ox 
jee ep wink 
and so on. None of the factors of the coefficient of X, in (6) vanish. 
For C (by hypothesis) is not zero. The equation, H, = 0, is virtually 
of the form (3), which we have supposed inadmissible. And, if h, 
were zero, we should have, by integrating the value of h,, 
BE ee fo COM ce cael iia na tl EN 
& being a constant quantity, that is, a quantity independent of p. 
But since X, and X, are not identical, there must be one factor of 
