106 
ON THE SURFACES THE LOCI OF 
12. It is to be noticed that, writing 
P = . hy — gz+aw, 
Q = — hx . +fz + bw, 
R= gx -fy . +cw, 
S = — ax — by — cz . , 
viz. P a = h a y — g a z 4- a a w, &c., then that we have identically 
\ , fi , v , p = — (\x 4- gy+vz + pw) . p^afiy, 
P a , Qa, Pa, S a 
Pp, Qp, Pp, &P 
Py , Qy , Py , $y 
and further that we have identically 
—p 2 a./3y = L a pP y + M a pQy + IS T a pR y + Q a p8 y , 
where 
L = (af' - a'f) x + (bf - bf) y + (c/ -cf)z- (be - b'c) w, 
M = (ag — a'g)x + (bg' — b'g) y + (eg' — eg) z — (ca — c'a) w, 
N = (ah’ — a'h)x + (bh' — b'h) y + (eh' — cli) z — (ab' — ab) w, 
n = (gb! - g'h) x + (hf - hf) y + (fg'-fg) 2 + (af - a'f + bg' - b'g + eh' - c'h) 
and L a p, &c. are the values of L, &c. on substituting therein (a a , ...) and (ap 
the unaccented and accented letters respectively. 
13. Observe that we have 
and similarly 
whence also 
L + (a'f+ b'g + c'h) x = . - c'Q + b'R -f'S, 
M+( „ ) y — c'P . — a'P — g'S, 
N+( „ ) z = — b'P + a’Q . -h'S, 
il + ( „ )w= f'P + g'Q + h'R . ; 
— L + (af + bg' + eh') x — . — cQ' + bP' —/S', 
-M + ( „ )y= cP' . -aR'-gS', 
-N + ( „ )z = -bP'+aQ' . -hS', 
— O + ( „ )w= fP' + gQ' + hR' . ; 
h'M - g'N + a'£l = - (a'f+ b’g + c’h) P', 
— h'L . +f'N+ b'il = —( „ )Q', 
g'L-f'M . +c'il = ~( „ )R', 
-a'L-b'M-c'N . =-( „ )S’]