THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 195 
Now the symbol 3’ written without a suffix being under- 
stood to belong to the first cycle (which is taken as a type 
of the rest), we may obtain =7° without developing the 
_ function on the right hand side of this equation, or finding 
the values of 7° corresponding to the other cycles. The 
following theorem enables us to pass directly from 72” or 
U? to Sr" or [U". 
29. Let 
T, or UF=2',(X,+X,++--+X,+X,), 
X being defined by 
Xn = Wut af ay a wet a a8 we vy ahacta’”’ &e., 
where a, 8, y, & and « are positive integers or (some of 
them) zero ; and m is the number of values of wv 2? aY a8 af. 
Then if we assume 
=X, = (a +a” +a” + &e.) Sat af vy x we, 
we shall have 
=r" or SU27= 3X, + osx, 7 Sri = ~SX,+ 3X, 
For, let > represent the sum of the 24 expressions for 7” 
or U", formed by applying all the cycles* (Art. 17) to any 
one of the values of r”” or U”. Then, since >’. X,, consists 
of 5, and consequently > >’,.X,, of 5-24, or 120, expres- 
sions of the form X,,, and since m= 120, or a sub-multiple 
of 120, it follows that 
Yr" or SU"= w0 eX. SX, + Oe “2 BX ,+ Ee 
But since 7” or U” is six-valued, we have Sr AS, 
and SU"=45U". Hence the theorem. 
30. Applying this : — Since (Art. 28) 
=>’ (walt Qatan.2,), 
Pa — i a a+ ae Sat By Ls, 
= 32a} 73+ 2Z22 2, 4,=0. 
* In Art. 17 only 12 cycles are exhibited, but the remaining 12 are giyen 
by reading the symbols in a reverse order, 
VOL, XV. DD 
