THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 183 
J (i) Sta, +P H+ P a+b a+? 2.=H7f(), 
then f'(#*)=?f(?), f(P)=#fE), and f@)=FFE)s 
~LOSLE) LE) LO=SO FE) SE) FE). 
That is, when /’(i), or if(i), is made the first factor, no 
new values of 7,(x) are introduced, but we are remitted to 
our former value. A similar argument holds for 7/(), 
#f(i), and zf(2). So that the number of values is still fur- 
ther reduced one-fifth, and 7,(#) has only 120+4-5 or 6 
values.* 
16. The same result may be arrived at by considering 
the form of the product, without reference to its factors. 
For 
THB, Ly +My Lg t+ XU tH, Lz+ Xs Xy 
is a circular function, each term being derived from the 
preceding by advancing the roots a step in a certain cycle 
++ Ly Vy Uy Ly Vs Ly Ly Vy Ly 5+ >> 
And, since the coefficients of the terms in the expression 
for tT are equal, we may, in forming its values, regard one 
of the roots, say 2,, as fixed, while the others are permuted 
inter se. Thus we shall have 1-2-3-4 cycles, giving rise 
to 24 corresponding expressions for 7; but since 
THX Ut Xo Ut %yUyt+ XX, 4232, 
HL Ly UX + Xe ky t Ly Xs t Ws Xs, 
these 24 expressions may be grouped in pairs, the members 
of each pair being equal. Hence 7 has only 12 values. 
Again : — Since 
T =H Let 23 ty Lat AU t+ U2, 
HL Ly t+ Ly Lyt LyX + UX + W522, 
the several values of 7 may be referred to the same 12 
cycles that arise in the formation of the values of r. Con- 
* For this proof I am indebted to my friend Mr. Cockle; and I may add 
that, having from time to time sent him portions of my researches, he has 
been so good as to give me the benefit of his more extensive reading on the 
subject, to point out coincidences, to suggest modifications, and in various 
other ways to help me to improye my methods and abridge my calculations. 
