ANI) LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 241 
of the third by £, and form a new ratio by addition of the numerators and deno 
minators. 
Then each of these ratios must be equal to 
«K ~ of + 2/3£ Q - a) (y — b) + y£ (y — b) 2 + 2 (x — a) + 2 h£ (y — b) + nl^“ + [&c (x — a) + ey(y — b )]- _ 
nJc£ n (ay— /3 2 ) + nk£ H ~ 1 (ae 2 y 3 — 2/3Sexy + yS 2 x 2 ) — £ (a/r — 2/3gh + yg 2 ) — (h8x — gey) 2 
Hence, by means of the equations f — 0, <f> = 0, each of the ratios must be 
equal to 
(n — 1) lc£ n /\_(n — l)k£ n (ay — /3 2 ) + (n — l)&£" -] (ae 2 y 2 — 2/38exy -f- yS 2 x 2 )] 
X.C. 
£/[£(ay — /3 2 ) + (a e 2 y 2 — 2/3Sexy -f- yS 2 X 2 )]. 
Hence it will be sufficient to prove 
(,r — a) [&c (x - «) + ey (y - &)] _ f _ 
lcQ l ~ l (ySx — /3ey) — h (h8x — gey) £(ay — (3 2 ) + (ae 2 y 2 — 2/3Sexy + yS 2 x 2 ) 
Now using the values of a; — a, y — b given above in (A), 
[Sx (x — a) + ey (y — 6)] [£ (ay — /3 3 ) + (ae 2 y 2 — 2/3Sexy + yS 2 x 2 )] 
= £ [&» (^ - gy) + ey (y/3 - ha)]. 
Hence, using the value of (x — a), it is necessary to prove that 
[(h/3 - gy)£ + ey (hhx - gey)] [Sx ( h/3 - gy) + ey (g/3 - lia)] 
= [(ay — /3 2 ) £ -|- (ae 2 y 2 — 2/3Sexy -f- yS 2 x 2 )] (ySx — /3ey) — h ( hBx — yey)]. 
Hence it is necessary to show that 
(ySx — f3ey) \k£, n (ay — /3 2 ) + kt, n ~ l (ae 2 y 2 — 2/3Sexy -f- yS 2 x 2 )] 
= £ [A (ay — / 3 3 ) (ASx — yey) — y (7?/3 — gy) (ySx - f3ey) + h (h/3 — gy) (/3Sx - aey)] 
+ (h Sx — yey) \]i (ae 2 y 2 — 2/3Sexy + yS 2 x 2 ) — yey (ySx — /3ey) -J- hey (/3Sx — aey)], 
i.e., 
(ySx — /3ey) (f> = 0. 
Hence this is satisfied. 
Hence the conditions for contact are satisfied. 
Since uv — w z = 0 is not satisfied at all points of the locus <f>= 0, the factor of 
the discriminant corresponding to it occurs only once. 
MDCCCXCII.—A. 2 I 
