87 
435] 
ON THE SIX COORDINATES OF A LINE. 
the equation of the hyperboloid is 
(agh) a? + (bhf) y 2 + (<fg)z 2 + {obc) w 2 
+ [( a bg) — {cah)] xw + [(bfg) + (chf)] yz 
+ i(^h) - {abf)] yw + [{cgh) + {afg)] zx 
In fact, we have 
(agh) = a 1 (gji 3 - g 3 h 2 ) + g i (Jim 3 - h 3 a 2 ) + h, {a 2 g 3 - a 3 g 2 ) 
= a. gh + g .ha + h. ag, 
where a, &c. stand for a 1} &c. and gh, &c. for g 2 h 3 — gJi 2 , &c. Hence the foregoing 
equation may be written 
oc 2 {a. gh + g . ha + h. ag) 
+ y* (b . hf 4-h.fb +f. bh ) 
+ z 2 (o-fg +f.gc +g.cf) 
+ w 2 {a .be + b . ca + c . ab) 
a.bg + b .ga + g.ab\ f b .fg+f. gb + g .bf\ 
\— c . ah — a .he — h. ca) y \+ c . hf + h .fc +f. eh) 
/ b . ch + c . hb — h.bc\ / c . gh + g . he 4- li. cg\ 
+ yW [- a.bf-b.fa -f.ab) + ZX \+a.fg+f.ga+g.af) 
' a.hf+h.fa+f. 
.+ b . gh + g. hb + h. v 
( c.af+a.fc +f.ca\ ( 
\— b. eg — c . gb — g .be ) y \- 
bc .w ( hy — gz + aw) 
4- zw 
55. This is 
+ ca . w {— hx +fz+bw) 
4- ab .w( gx —fy + cw) 
+ gh. x (ax + by + cz) 
+ hf .y (ax + by + cz) 
+ fg. z (ax + by 4- cz) 
+ af [w (ax + by + cz) — x{ hy — gz 4- aw)) 
4- bg [w (ax + by 4- cz) — y (— hx +fz 4- bw)] 
4- ch [w{ax + by + cz)-z{ gx -fy + cw)) 
— bf.y { hy-gz + aw) 
— cf.z { hy-gz+aw) 
— eg . z (—hx +fz + bw) 
— ag. x {—hx +fz + bw) 
— ah. x {gx —fy +cw) 
— bh.y {gx — fy 4- cw) = 0.