RESOLUTION OF ALGEBRAICAL EQUATIONS. 135 
there will be ss; ......... groups of such terms as (3). Still further, 
without having respect to the surds ¢, ¢, , &c., there may be (Cor. 5, 
Prop. VI.: see more particularly the explanation presently to be 
given) m distinct groups such as (8): only (as has been proved) the 
gar functions in (8) are the only unequal terms in all the m groups 
On the whole, the series of cognate functions of f (p), taken on a non- 
recognition of the surd character of those surds alone which are pre- 
sent in F, will embrace mnvrss, ......... terms, or mss, ......... lines 
of terms such as (8), of which the following may serve as examples: 
Py» Py Joes ter ne nee ang Pur? 
v, p) Wh Qiisiieele whe Oo =) a) 5 Yee 
yi ' ' ‘ soe cesceae 8+ eae (4) 
PE Ste gt ANS eB cn sa be eae 
lic oll ale gal aaa al 
The first of these lines is (3). The second is a cluster of terms, in 
addition to the mr terms of the first line, obtained without having res- 
pect to ¢, ¢,, &c., and being a repetition of the values of the terms 
in the first line; for, in the mnr terms, obtained without reference to 
t, & , &c., the unequal terms which constitute the series (3) are all 
repeated (Cor. 5. Prop. VI.) the same number of times. The third 
line of (4) contains the terms in the first line, transformed by chang- 
ing ¢ into 2 ¢; % being an st root of unity, distinct from unity. 
And those in the last line contain the terms of the second line, trans- 
formed by a similar change of ¢ into 2, ¢. Now it can be shown that 
the terms of the third line are equal, in some order, to those of the 
first, each to each. For, since ¢, present in F, (x), disappears from 
F, it follows that the continued product of the factors of F, viz. : 
F, («), Fy («), &c., remains the same when 7, ¢ is substituted for 7. 
That is, the factors, 
G— oy, x — Po Dictelsdteitrobol sh tints o— Pur, 
are the same, taken in some order, with the factors, 
®— pe » & — i Jiteeseverse-9 U—O 
Hence the terms in the third line of (4) are, in some order, equal to 
those in the first line, each to each. In the same way it may be 
proved that all the mrss, ......... cognate functions above described, 
are merely repetitions of the values of the functions in (8). Hence 
the terms in (3) are all the unequal cognate functions of f (p), obtain- 
