RESOLUTION OF ALGEBRAICAL EQUATIONS. 14] 
Hence, by a comparison of (6) and (7), 
I(h-h, ) hy h 
1—z, =0 ois h=hy oe Vv, = Vi. 
ae 
Hence be a vi becomes 62 V V,: which, again, by returning 
kh-ws Ih-w4s i 
from V and V, to Y and U, becomes 6 Y LUE ; where ws is 
the greatest multiple of s in kh, and ws is the greatest multiple 
of sin ZA; 64 being an expression clear of the surds Y and U. Con- 
sequently F, (a) may be written, 
kh-ws jh-w,s 
Gs BNC n We ee a Mo (3) 
B. being an expression clear of the surds Y and U; and the 
numbers, 4, J, pene the same in all the terms, such as 
py? gh ot ineiided wader the symbol 3. But equation (8) 
implies that the surds Y and U are similarly involved in ¥, (a). 
Proposition X. 
Let f (p) be an integral function of a variable p, in a simple 
form, satisfying the conditions of Def. 8; end let 
Ae AB Us otelag 4 \(1.) 
where Y isa surd in f(p), with the index =; and A is an expres- 
sion, not zero, involving only surds, distinct ee Y, which occur in 
JS (p); » being a whole number, not zero, and less than s; and the 
expressions B, U, are what A and Y respectively become on changing 
T, a chief subordinate of Y, with the index —, into z T, 2 being a 
; o 
ot root of unity, distinct from unity; T not being a subordinate of 
any surd in the expression A. Then the surd Y is of the form, 
Yat TAR) eset sit 
where H is an expression clear of the surd T; and m is a whole 
number, less than oc. Also, o is not equal to s. 
For, let » be the general expression which includes all the cognate 
i 
functions of A Y , taken without reference to the surd character of 
any of the surds in A y , except T and Y; and let d> arranged so as 
to satisfy the conditions of Def. 8, be, — 
Vou. V. M 
