THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 185 
corresponding respectively to the cycles 
Ty x4 & 
i a eee - -_——— 
ae oe a ms, oe a 
*4 Me Re. my igs yy Ee 
Mi i ‘2 Si 4 ‘3 Ti ¥ 3 
H Fi ! 1 8 : 9 ' 
1 i 1 ! ' t 
\ , \ / \ : 
\ £ re y \ / 
x 7 Si oft x vs 
‘2 Sak. SB Xs Z5 = eee c U4. Lz, we na Ly 
L Wy x 
woot aen wes pee 
t ’. a ~. - ~ 
Ny: ¥e KA s of xX 
/ 
\ y <\ ‘ 
Wy 4 U3} U3 aie 4 
1 I 1 
1 1 
Tho setitorce", Ve fare nyo heehee’ Py gt8 Sod 
‘ ! 
4 / \ Fi \ ‘s 
*y “f \ ff ‘ / 
4 4 
a XN. ve 8. oe 
- ~~. -- etn he 
X~ ———— ee AL Uy ~———— Xs X3 ----" Ig 
18. New Cyclical Symbol. Let (0) be a function of a, 
and let x(qg) be derived from y(0), by advancing each of 
the roots contained in it g steps in the cycle r. And let 
=x (0) =x(0) +x(1) +x(2) +x(3) +x (4). 
Then 
2, R= Da", >, vt (et +22) = sar zt, &e., 
and 
‘ Pay ae = = er ta 
1% = 472%, 10 %3= 247% #3, 
2/2 2, Hy= Dg 2, Xe, 2, 2 U3= 2/5 2, 25, 
, ‘4 / 
B's By Pg D9 Ly Ls 5 i's Ly Lg= Dig 2 Bs y 
D4 Py Lg= DX, X25 why 24= > oP, X45 
, / 
D's Ly Py= 2X Ls, D5 By Lg= DX Ws, 
Big ty Ba 2 lly oe Do fia el Lew 
The six values of r7’ may now be briefly expressed as 
follows : 
Ty T = By Ly BL ye= By Uy 7 Uy Ls=T, 7 
Tz T g= Dy Vy Ly Dy Vy Le=D"g Ly Ly B'gL, Lz=Ty T's 
737 g= 3X) Ly % gL Ly= Dg Ly Uy X'g Lj Lz=Ty T'g, 
TT gi" Hy Wy D4 Ly MH QV Wy 9X, He=TyT'0, 
TT p= D5 By Ly 5X, Ve= JV, My BX Le=TT yn, 
ToT GD GX Lyd GX, MHD py Hy Dy M=T yy. 
>’, must not be confounded with Mr. Cockle’s >’ of 
