150 RESOLUTION OF ALGEBRAICAL EQUATIONS. 
is not of less dimensions than the H. C. M. of F, () and any term 
in the first line of (2). The fact, therefore, of R being zero, implies 
that F, () has a common measure with each of the terms in the first 
line of (2), and that its H. C. M. with any of these terms is of the 
same dimensions as its H.C. M. with any of the rest. Let L be 
the H. C. M. of F, (~) and 1X, , Ly that of F, («) and 2X), and so 
on. The terms L, Ly ,.&c., are all of the same dimensions; L, , in 
fact, being what L becomes on substituting 2 Y, for Y, ; and so on. 
Also, since all the factors of the terms in the first line of (2), being 
factors of F.+ («), are unequal, it follows that all the factors of the 
terms L, L, &c., are unequal. This, taken in connection with the 
fact that F, () is a factor of the continued product of the terms in 
the first line of (2), shows that F, (#) is equal to the continued pro- 
duct of the terms 1, L,, &c. Thus the first of equations (9) is 
established. In the same manner the others can be established. 
Cor. 4.—Should F, («) not be equal to a term in (2), an equation 
of the form, 
B A 
Dh ABET e ag Mee tthe ale eee ie NTP) 
must subsist ; 8 and \ being whole numbers, distinct from zero: and 
P an expression involving only surds which occur in F,(), exclusive 
of Y; while Y, is what Y becomes when T is changed into 2] . 
For, let N be the H. C. M. of *F, (x) and ,X,, Nj that of on (x) 
and »X;, and so on, Q being the H. C. M. of a (x) and 1X, 
Q, that of "BR, (x) and yX5, and so on: in which case the terms, N, 
Ni, &c., are respectively what L, Li, &c., (see Cor. 3), become on 
changing Y into zY; and Q, Q,, &c., are what K, K,, &c., become 
on changing Y into z Y. Then, in the same way in which equations 
(9) were found, we can establish the equations. 
*B,.(@) = N x Ny x Ne xen he x No 
TO) helen Gh Se Op ea x Oe 
Now suppose, if possible, that such an equation as (10) cannot subsist. 
‘Then, exactly as it was shewn in Cor. 2, [proceeding upon the hypo- 
thesis that such an equation as (8) cannot subsist], that any function 
involving merely such surds as are in F, («), together with Y; , is m 
a simple form, we may demonstrate [proceeding upon the hypothesis - 
