DE. PLUCKEE ON A NEW GEOMETER OE SPACE. 
759 
Finally, by putting the values of r and s taken from the last two equations into the 
first one, we obtain 
{(B' +Yx -D'*)E"-F'Ify}Ea* 
+ { ( A" — Y'y — WzJD' — E"F'^ } Dyz 
-}- { ( A" — F"?/ — F'z) (B' + Y'x — D'z) + F'F"xy } (C + 1>/ + E#) =0, 
which, by the disappearance of terms of the third order, becomes 
A"B'C+A ,, (B'E+CF)^+B , (A"D-CF")3/-C(A"D'+E"B'>' 
+ A'T'Etf 2 — B'F"D^ 2 + CE'TO 
+ (A"FD-B , F , E)^-(A"D'E+CE"F)^ I - * ( 91 ) 
+(CF"D'-B'E"D)y2=0. j 
After dividing by A"B'C and replacing 
E 3) IP _F E" F 
C’ C 5 B'’ B'’ A"’ A" 
by I, ?i, £', l', 71 ", the last equation assumes the following symmetrical form, 
+1 fit'+wy+w** | (92) 
+(^+iv')^+(r?"+r)«+w+>/'?v=o.| 
In order to represent the configuration this equation replaces the three equations (90), 
which may be written thus, 
vr+!g — 1 = 0, 
£' <r -i'(s£-rff)-l=0,l (93) 
?e-A*s-n)+ 1=0- j 
It shows that the configuration is a hyperboloid touching the three planes XY, XZ, 
YZ. The rays within these planes are represented by 
2=0, lx +7iy =1,1 
y= 0 , r*+r*=i,| (94) 
#=0, >/ty+£"2=l,j 
the directrices within them by 
2=0, g'jF+V'y =1,1 
y= 0, gar +$"*=1,1 : (95) 
*=0, ny+?z= 1.1 
The points of contact, being within each plane the intersection of the ray and the 
directrix, are easily obtained. 
The rays within the three planes of coordinates which form one edge of a circum- 
scribed parallelopiped meet the directrices within the planes forming the opposite edge. 
5 m 2 
