IN PARTICULAR THOSE OE A QUADRIC SURFACE. 
167 
the curve 
[508 
508] 
(<r an arbitrary parameter) as the equations of a right line on the surface; viz. 
considering x, y, z as denoting the foregoing functions of p and q, these two equations 
are forms of a single relation between p, q, a, which relation expresses that the point 
(p, q) is situate on the right line determined by the parameter er. We may from 
this integral equation deduce without difficulty the foregoing differential equation; viz. 
we have 
dx idy 1 dz 
fa \Jb a fc ’ 
or multiplying these equations, 
dx 2 dy 2 dz 2 
1—— 
a b 
and substituting herein for dx, dy, dz their values in terms of dp and dq, we find 
the required equation 
Supposing 
dp 2 dq 2 
(a+p)(b+p)(c+p) (a + q)(b + q)(c + q) 
18. I return to the integral equation involving <r : we have to rationalise this 
equation, that is, obtain from it an equation containing x 2 , y 2 , z 2 , and then substituting 
ature, the 
for these their values in terms of p, q, we have the required relation between p, q, a. 
e, in fact, 
'on is not 
eodesic is 
We at once obtain 
however, 
or if for greater convenience we introduce in place of a a new parameter <£, deter- 
1 2(6 
mined by the equation a 2 + — = — , the equation is 
3 surface ; 
lifferential 
Writing for shortness p + q = X, pq= Y, we have 
-ßy- 2 =a 2 + aX+Y, -ya^ = b 2 + bX+Y, -aß-=c 2 + cX+Y- 
' a b c 
the right 
and substituting these values, the equation becomes 
{/3 (b 2 + bX + F) — a (a 2 + aX + F) — <j>y (a/3 - c 2 - cZ - Y)} 2 + 4a/3 (0 2 - f) (c 2 + cX+Y) = 0, 
or, what is the same thing, 
{¡3b 2 — a.a 2 — (f)7 (a/3 — c 2 ) + X (/3b — a.a + 4>yc) + Y(¡3 — a + <£y)} 2 + 4a/3 ($ 2 — <y 2 ) (c 2 + cX + 1 ) = 0. 
19. This is an equation, quadric as regards p, and also as regards q, viz. it is 
of the form 
(a + 2hp + gp 2 ) 
+ 2q (h + 2bp + ip 2 ) 
+ q 2 (g+ Zip + cp 2 ) = 0,