•286 ME. A. CAYLEY’S SEVENTH MEMOIE ON QU ANTICS. 
duced in the place of PU, QU ; and from the expressions so transformed are deduced 
also the expressions for Y(aU-l-6j3HU), Z(o5U + 6|3HU). Table 75 is to be deduced 
from Table 69 by writing therein (a', jS') for (a, j3), and then putting da'PU+jS'QU 
= 2aYU — 2j3ZU, which, as is seen above, gives a', (3' as functions of a, |3 and of the 
iuA ariants S and T ; but in some of the formulae YU, ZU, have to be introduced in the 
place of PU, QU. And so for the Tables 76 and 77. The actual etfectuation of the 
transformations would, it is almost needless to remark, be very laborious, but th© forms 
of the results are easily foreseen, and the results can then be verified by means of one 
or two coefficients only. The new Tables are 
Table No. 74. 
R(aU+6i3HU)=Rx[(l, 0, -24S, . . (3y]\ 
S («U+6i3HU)= (S, T , . . Xoi, /3)^ 
T(aU+6/3T[U)= [(T, 96S^ /3)J. 
(aU + 6/3H U) = aU + 6/3HU, 
H(«U+6(3HU)=— ^Xf B,(l, 0, ^248, . .J_cc, I3y .V 
1 -63.(1, 0, -24S, |3)‘.HU, 
C(<2U+6,SHU)= (1, 0, -24S, /3)‘x 
f 3,(S,T, 
1-63,(S,T, ..J,., py.JlU, 
D(e<U+6/3HU)= 0, -24S, . . la, (3)‘x 
f 3,(T, 96S“, 
1-63.(T, 96S’, (3)' -HU, 
P(.U+6/3HU)=-5Lxj sat, ..I«,/3r.YU 
1+3.(S,T, ..5;«.|3)‘.ZU, 
Q(»U+6/3HU)=-ixi S,(T. 96S»,..5;»,/3)».YU 
l+3.(T, 96SS..Ic<,/3)».ZU, 
Y(aU+6(3HU)= [(1, 0, -24S, . . /3)‘]=x 
(-2kYU+2/3ZU), 
i[(i, 0, -24S, OTx 
r 3,(1, 0, -24S, (3)<.YU 
1 + 3 .( 1 , 0 , -2iS,..%a,fiy.ZV. 
Z(aU+6(3HU)= 
