AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 201 
or, putting x — a = X/Z, y — b = Y/Z, 
aX 2 - 2 n (xZ + X) (£Z + xX - y Y) 
24:nfix- [9a/3£ + (/3Y 5 + ay 3 )] 
_ /3Y 2 + 2?i (yZ + Y) (gZ + .rX - yY) 
— 24nocy 2 [9a/3£ — 4?i (/3x i3 + at/ 3 )] 
- 2tiZ (gZ + xX - y Y) 
18a/3 [9a/3f + 4ti (/3Y 5 — ay 3 )] 
It will now be shown that the values of x, y, z which satisfy (95) also satisfy 
these equations. 
For substituting from (95), these equations become 
aX 2 + 4?i£Z (xZ + X) 
Anfix- [9a/3f + An(fix s + at/ 3 )] 
These reduce further to 
/3Y 2 - 4 n£Z (yZ 4- Y) 
— 4 na.y~ [9a/3£ — 4n (fix* + ay 3 )] 
_ 4n£Z 2 _ 
oa.fi [9a/3£ + 4ti (/3» 3 — ay 3 )] 
i.e., 
_ 4?t£ZX _ _ — 4n£'ZY 
Anfix 1 [9«/8C + An(fix* + «y 3 )] — 4?iay 2 [9a/3£ — 4?t (/3.x 3 + ay 3 )] 
__UtfZ 2 _ 
3a/3 [9a/3£" + 4?i (/3a;- 3 — ay 3 )] ’ 
_X_Y_ 
Anfix" [9a/3£ + 4 n (fix 3 + ay 3 )] 4?tay 2 [9a/3f — 4?i (/3ar 3 + ay 3 )] 
_ Z_ 
3a/3 [9a/3£ + 4 n (fix* — ay 3 )] 
Now the relations (95) satisfy (94). 
Further (94) can be written in either of the forms 
[9a/3£ + 4 n (fix 3 -f a 2/ 3 )]' = 1 AAnccfi^y 3 , 
[9a/3£ — An (fix 3 -f a y z )J — ~ lAAnafi-^x 3 . 
Hence it is necessary to show that 
_X_Y__ Z_ 
48 nafix 2 y \/(nfiy£) 48 nafixy 2 \/( — notx£) 2An»fixy ^/(— afixy) 
i.e., 
X 2 /Z 2 = — Axilla. 
Y 2 /Z 3 = 4 nylffi, 
and these are true by (95). 
Further, <£ = 0, _/= 0 are both consequences of (95). 
Hence all the equations (96) are satisfied by the same values of x, y, z. 
MDCCCXCII.—A. 2 D 
