RESOLUTION OF ALGEBRAICAL EQUATIONS. 15] 
that such an equation as (10) cannot subsist] that any function invol- 
ving merely such surds as are in F, (x), together with Y; and Y2, is 
in a simple form. This being premised, we remark, that, in (9), L is 
either equal to one of the expressions K, K,, &c., or has a common 
measure with more than one of them. Let K, be a term in the se- 
ries K, K,, &c., such that the H. C. M. of L and K, is not of less 
dimensions than the H. C. M. of L and any other term in the series. 
Take K, the general form which includes all the terms K, K,, ...... 3 
K, , and likewise all the terms Q, Q,, ...... , Qs-1; the latter series 
being derived from the former by changing Y into 2 Y. Perform 
the operation of finding the H. C. M. of L. and K, stopping at the 
point where, in the particular case of L and K,, the process comes to 
an end. If at this stage the remainder be R, and R; be the corres- 
ponding remainder in the case of L and K,, the forms of R and 
Rj are, 
Au Aah aba 
ESS call VD +qY Y, Yo ida tent 
I 
ae 
8 
ic>| 
-— 
E AAC ACS 
er = oe +x ( iia +q2Y x Me if wy) + &e.; 
the expressions being similar to those in Cor. 3. But since R; = 0, 
we find (as in Cor. 3) that R = 0; it being kept in view that any 
function which involves merely such surds as occur in F, (*), together 
with Yj and Y;, isin asimple form. Hence L has a common mea- 
sure with every term included under the general symbol K, and 
therefore it is a factor of °F, (=) as well as of F, (): which, since 
F. («) and °F, (2) have no common factors, is impossible. Therefore 
an equation such as (10) must subsist. 
Cor. 5.—The same suppositions being made as in Cor. 4, the fol- 
lowing equations must subsist : 
NeW gg PY 4 | 
B° A,—B 
YY, a Py Yo 3c) 
B° Ag—BA | 
yy pie a tian BL Rebs Ty LY) 
gt pe ey 
YYg = Ps Y4 ; 
B° A4—BAg 
Y Yio = Ps Y5 j 
