129] 
AND ON AN IRRATIONAL TRANSFORMATION &C. 
149 
Recapitulating, l, m, n are considered as functions of x x , y x , z x determined (to a 
common factor pres) by the equations 
lx x + my x + nz x = 0, 
l 2 bc + m 2 ca + n 2 ab = 0 ; 
© is determined as above, and then writing 
a = ax x 2 + by 2 + cz 2 — a (x x 2 + у 2 + z x ), 
/3 — ax x 2 + by 2 + cz 2 — b (x x 2 + y x 2 + z x ), 
7 = ax x 2 + by x 2 + cz x - c (x x 2 + y x 2 + z x 2 ), 
we have 
x 2 = ®y x z x (/3n 2 + 7m 2 ), 
у-i = ®z x x x (71 2 + an 2 ), 
z 2 = ®x x y x (am 2 + ¡3l 2 ); 
and these values give 
lx 2 + my 2 + nz 2 = 0, 
X? + yi + Z 2 = X 2 + y x 2 + z x 2 , 
ax£ + by 2 + cz 2 = ax x 2 + by 2 + cz x 2 . 
In connexion with the subject I may add the following transformation, viz. if 
3 Va x' = V3/3 (y — z) + V(3a — 2¡3) (x 2 + y 2 + z 2 ) + 2/3 {yz + zx + xy), 
then reciprocally 
3\//3 x — — V3a (y' — z') + V (3/3 — 2a) (x' 2 + y' 2 + z' 2 ) + 2a (у V + z'x' + x'y'). 
x 2 + y 2 + z 2 = x' 2 + y' 2 + z' 2 , 
¡3 (x 2 + y 2 + z 2 — yz — zx — xy) = a (x' 2 + у 2 + z' 2 — yz — z'x' — x'y'). 
Suppose 1 + p + p 2 = 0, then 
ж 2 + y 2 + z 2 — yz — zx — xy = (x + py 4- p 2 z) (x + p 2 + pz) ; 
and in fact 
3 Va {x' + py' + p 2 z') — — V3/3 (1 + 2p)(x + py + p 2 z), 
3Va (x' + p 2 y'+ pz) = V3/3 (1 + 2p) (x + p 2 y + pz). 
The preceding investigations have been in my possession for about eighteen months. 
2 Stone Buildings, April 18, 1855.