284 ME. A. CATLET’S SEVENTH MEMOIE ON QHANTICS. 
and the two derived contravariants, 
YU= 3T.PU-4S.QU, 
ZU=:-48S^PU+ T.QU, 
which for the canonical form are as follows : — 
Table No. 70 (addition to). 
CU=(l + 8/^) [(_l+4/^)(^‘®+/+^^)+ m wyz'], 
DU=( 14 - 8 /®) \P (5+4/^)(a;®+y+2!®)+3(l — 
YU=z(l + 8^7[ /(r+;?=’+r) -3f<], 
ZU =(1 + 8^^)^[(l+2Z*)(r+;j*+n+18^W]- 
244. Y'e have, conversely, 
3R. U=3T .CU+24S.DU, 
3P.HU=8S^CU+ T.DU, 
and 
- 3R . PU = T . YU + 4S . QU, 
-3R.QU=48S^YU^-3T.PU, 
and also the following formulae, viz. if 
2aCU - 2/3DU = a'U + G/S'HU ; 
then 
a'=-2Ta-16S^/3, 
^’= 8Sa+ T/3, 
which give, conversely, 
a=X( t,«'+16S“/3% 
/3=^(-8S»'- 2T/3'); 
and moreover, if 
- 2aYU+ 2/3ZU= 6a'PU -f /3'QU, 
then 
a'= — ( Ta+16S^|S), 
/3'=-(-8Sa- 2T^), 
which give, conversely, 
« = -4(-2Ta'-16S*(3'), 
(3=-i( 8S»'+ T/3'); 
SO that the relation between (a, /3) and («', |3') in the present case is similar to that 
between [a, f3') and (a, f3) in the former case. It may be noticed that in all these 
systems of linear equations, the determinant of transformation is a multiple of 
64S*-P( = R). 
