420 
ME. A. CAYLEY’S MEMOIE OX CLEATIS OF THE THIRD OEDEE. 
We have also, identically, 
ABC-FGH=:y(-^+?^)XYZ{Z^(X*+Y^+Z^)-(l+2Z^)XYZ}, 
which agrees with the relation ABC— FGH=0. 
7. Before going further, it will be convenient to investigate ceitain relations which 
exist between the quantities (X, Y, Z), (X', Y', Z'), connected as before by the equations 
XX'+Z(YZ'+Y'Z) =0, 
YY'+/(ZX+ZX)=0, 
ZZ' +Z(XY'H-X'Y)=0, 
and the quantities 
I = YZ' - Y'Z, a:=XX'= -7(YZ' + Y'Z), 
=ZX -ZX, |3= YY' = -)(ZX' +ZX), 
^=XY'-X'Y, y=:ZZ' =-7(XY'+XY). 
AVe have identically, 
2XX'(YZ'-Y'Z)+(XY'+X'Y)(ZX'-ZX)+(ZX'+ZX)(XY'-XY)=0; 
or expressing in terms of |, a, j3, y the quantities which enter into this equation, 
and forming the analogous equations, we have 
2^a| — yri — j3^ =0 (A) 
— yl + 2 ^/ 3;?— =0 
— j3| — af} -j-2^7^=0. 
We have also 
XW'Z'-X'WZ=i{-(XY'+XY)(ZX'-ZX)H-(ZX'+ZX)(XY'-XY')}, 
and thence in like manner, 
XWZ' -XWZ~(y, -(3Q (B) 
Y^Z' X'- Y'^ZX ) 
ZX'Y' -XWZ=^^(f3| -ari). 
Again, we have 
(YZ'-Y'Z)^=(YZ'+Y'Z)^— 4YY'ZZ' 
(ZX'-ZX)(XY'-X'Y)=-(ZX'+Z'X)(XY'4-X'Y)+2XX'(YZ'+Y'Z}; 
and thence 
?=l.a*-4(3y 
(C) 
