DR. PLUCKER ON A NEW GEOMETRY OE SPACE. 741 
The following theorem is the geometrical interpretation of the equation (25). 
Draw through a point P its corresponding plane _p, and the plane XY perpendicular 
to the axis of the complex meeting that axis in O. Let P be an arbitrary point ofy>, 
and B' its projection on XY. The double area of the triangle POP' divided by P'P is 
a constant, and equal to Jc. 
33. In order to generalize, we may start from the equation 
Ar+Bs+C-t-D<r+Eg-j-F(sg — ra)— 0 . (7), 
and proceed in the following way. By replacing x, y, z by f, ?j, S- (see [16], note), 
and omitting the accents, we immediately derive from equation (10), 
|= C u — By — D w, 
j? = — Ct -p Aw -}-Ew, (27) 
£= Bt —Au—Fw, 
F)t— Fu+Fv, 
|, *i, £, S indicating any point, and t, u, v, w its corresponding plane. From the first 
three of these equations results the equation 
A| + Br, + C£ = — ( AD — BE + CF) w, 
which, multiplied member by member by the fourth equation, 
F)t— Em+Fw=S-, 
and divided by Sw, furnishes the following relation, 
(AH-By+Cz)(D^-E2+F^) =— (AD— BE+CF). . . . (28) 
In a similar way we obtain 
(C3 + E£ + D>)) 
Bt—Au—Fw 
? 
V 
B3-D$ + F£ 
— C t -f- A u -f- 
>1 
u 
> 
ip 
1 
1 
Cm — Bv — Dw 
— £ t 
= — (AD— BE+CF). } 
34. In starting again from the equation (26), 
sg — r<r=Jc, 
and in supposing that there is a right line determined by means of the coordinates of 
any two of its points (ad, y\ z') and (#", y", z") according to [31], its conjugate line will 
be represented by the system of equations, 
y’x — ady~Tc{z — z ' ), 
y"x-x"y=Jc(z—z"), 
5 K 
MDCCCLXY. 
