AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 253 
and substituting in the first of equations (/3), after making some reductions, the 
result is 
s 2 
——fffi ( 22 + 2a&m — m 2 ) [2 abm — ( z 2 — m 2 ) (z — 1)] = 0, 
which is satisfied by (a). 
Hence the first of equations (/3) is satisfied by the values of x, y, z given by (a). 
Again substituting for x, y in terms of 2 from (a) in (y) the ratios become equal to 
[z s (— m — 1) — m 2 z 2 + (m 2 — 2 abm) z] / m (m + z) 
= [z 3 (— m + 1) + m 2 z 2 — (m 2 — 2 abm) z\ / m(m — z) 
— [32 4 — 22 s + $ (2 mob — 3m 2 ) + 2 (2 m 2 — 4 abm)] / m ( m 2 — z~). 
Hence it is necessary to show that 
[z 2 (— m — 1) — m 2 z -f (m 2 — 2 abm)~\ (m — z) 
= [ 2 2 ( — m + 1) + m 2 z — (m 3 — 2 abm)] (m + 2 ) 
= 32 s — 22 s + z(2mab — 3m 2 ) + (2m 3 — 4 abm) .... (S). 
Equating the first and second quantities in ( 8 ) it is necessary to prove that 
2 3 — z 2 — m 2 z + m 2 — 2 abm = 0, 
which holds by (a). 
Equating the second and third quantities in ( 8 ) and removing the factor (m + 2 ), 
the same result is obtained. 
Hence the values of x, y, z given in (a) satisfy all the equations (/3), (y). 
Art. 23 .— If the parameters of one of the two Surfaces having Conic Nodes become 
infinite, and if C = 0 be the equation of the Conic Node Locus, A contains C 3 as 
a factor. 
The conditions that one value of a and one value of b satisfying the parametric 
quadratics (180) and (181) should be infinite are that 
4 (to _w*) = o, |(« 1 .-w») = 0 , l(uv- w*) = o. 
In this case the values of A and dA/dx, as given by (146) and (147), both vanish. 
Hence A contains C 3 as a factor. 
Example 14. — Locus of one Conic Node. 
Let the surfaces be 
[a {x — a) + (3 (y — 8)] 3 + 2 gz (x — a) + 2 hz (y — b) + kz 2 = 0. 
