430 
ON THE COMPLEX OF LINES 
[859 
Collecting the terms multiplied by P, Q, R, ®, respectively, we have 
il = P {;ya8 3 — x/38 3 + Sway 3 + xy 3 8 — yy 4 } 
4- Q {— 3ya(38 2 + Sza 2 y8 — Swa 2 y 2 + Swa/3 2 8 — Sxay 2 8 + 3x/3' 2 8 2 } 
+ R {— za 3 8 + iva 3 y — wa(3 3 — 3x/3 3 8 + z/3 4 } 
4 ® {— 2 yza 2 8 2 — zwa 2 (38 + 2 xza/38' 2 4- 2ywa 2 y8 — xyay8 2 
4 w 2 a 2 /3y — 4z 2 a/3 2 8 + (— 2xw 4- 8yz) a/3y8 — 4y"ay 2 + x 2 j3y8 2 
4- zwa{3 2 y — 5ywa/3y8 — 5xz(3' 2 y8 + xy(3y 2 8 
— w' 2 /3 4 + \z 2 (3 3 y + (2xw — Qyz) /3' 2 y 2 4- ^!f(3y 3 — x 2 y 4 ], 
which may be written as follows:— 
O = P [y (a8 3 -y 4 )-\-x (y 3 8 — /38 3 )} + P {Sway 3 ) 
4- Q [Sx (/3 2 8 2 — aY 2 S) 4- 3w (a(3‘ 2 8 — a 2 Y 2 )} 4- Q (Sza 2 y8 — 3yoc/38 2 ) 
+ R {— z (a 3 8 — /3 4 ) + w (a 3 y — a/3 3 )} 
4 © \zw (— a 2 /38 + a8 2 y) 
4- xz 2 (a/38 2 — (3 2 y8) 
4 yw 2 (a 2 YS — a/3y 2 ) 
+ xy (— ay8 2 + /3y 2 8) 
+ xw 2 (— a/3y8 + /3 2 y 2 ) 
4 yz (—2a 2 8 2 + 8a/3y8 — 6/3 2 y 2 ) 
+ x 2 (/3y8 2 — 7 4 ) 
4- y 2 4 (— aY 2 S + /3y 3 ) 
+ ^ 2 4 (— a/3 2 8 + /3 3 y) 
4- w 2 (a?(3y — ¡3 4 ) 
+ R (— Sx/3 3 8) 
4- © (— 3xz/3 2 y8) 
4 © (— Sywa/3y 2 ) 
in which all the terms contained in the { } admit of expression in terms of 
P 0 , Q 0 , R 0 , © 0 ; the remaining six terms not included within { } may be written 
3wPa (y 3 — /38 2 ) 4- 3 (wP — yQ) a/38 2 — S®xz/3 2 y8, 
— SxR8 ({3 3 — a 2 y) 4 3 (—xR 4 zQ) o?y8 — 3%ywa(3y 2 ; 
which, observing that wP — yQ = xz%, and — xR 4- zQ = yw©, are 
— 3wPa (y 3 — ¡38 2 ) 4- 3iK2© (a/38 2 — /3 2 y8), 
— 3xR8 ((3 s — a 2 y ) 4- 3yw® (a 2 y8 — a/3y 2 ).