I 
372] ON THE RECIPROCATION OF A QUARTIC DEVELOPABLE, 
and writing □ =0, the values each divided by 3, are simply 
509 
& , 
XY, 
XZ, 
xw, 
YX , 
Y 2 , 
YZ, 
YW, 
, 
ZY, 
Z 2 , 
zw, 
wx, 
WY, 
wz, 
If 2 , 
equations in 
fact, 
becomes 
X8x + 
Z8z + W8w = 
or multiplying by 3, and attending to the equations (2), this is 
— 3 w8x + z8y — y8z + 3x8w = 0. 
This should be a consequence of the equations 
8 (3xz — y 1 ) = 0, 8(yz — 9xw) = 0, 8 (3yw — z' 2 ) = 0, 
that is, we should be able from the first three to deduce the fourth equation in the 
system 
3z 8x — 2у 8y + 3x 8z = 0, 
— 9 w 8x + z 8y + у 8z — 9x 8w = 0, 
3w 8y —2z 8z + 3у 8w = 0, 
— 3w8x+ z8y — у 8z + 3x 8w = 0, 
or we ought to have 
3 z, 
3x, 
— 9 w, 
y> 
— 9x 
3 w, 
-2z, 
Ь 
— 3 w, 
z, 
3x 
= 0 
but expanding, this is 
or 
6 (81 xhu‘ 2 — 54txyzw + 12xz 3 + 12 y 3 w — 3 y 2 z 2 ) = 0, 
(yz — 9 xw) 2 — 4 (xz — y 2 ) (yw — z 2 ) = 0, 
which is true in virtue of the relations (4). Or what is the same thing, we may show 
without difficulty that the equation 
— 3w8x + z8y - y8z + 3x8w = 0, 
is satisfied by writing therein x : y \ z \ w — \ : 6 : & 2 : ^ 6 3 .