284 PliOFESSOR C. J. JOLY OX QUATEIiXIOXS AXD PROJECTIVE GEOMETRY. 
If c Is a centre of generalized curvature, or a point at which consecutive normals 
Intersect, we liave for intersecting normals 
^ dc = dr/ + dy; + ]b~^p d^ = cdi«. . . (373), 
where An is some small scalar, and dc = cAu, because on the hypothesis that 
consecutive normals intersect in c, c and dc represent the same point and differ onlv 
in weight. On elimination of c, (373) becomes 
dry + yA'i dy) + v (ry-j- tlr^p) + ?(vy = 0, {tv = Af — t Au, v + w — — d?r) . (374); 
and as this may be written 
{lA-th 1/) (dry + z.vy) + ?cty = 0 , or dry + rr/ + u-(1 + ^//,-(d)-hy = 0 . (375), 
we find, on operating by S/9 or S/y, the equation 
Sy/(1 + = 0 .(376). 
On inversion of the function this ])ecomes a quadratic in t whose roots determine 
the two centres of curvature. 
91. This equation may be thrown into the more suggestive form'^ 
(/"^ + = 0 .(377), 
which shows that the roots t are the parameters of two of the quadrics of the 
singly infinite system Sr (/-i + r = 0, which pass through the point ry. The 
third quadric of the system through that point is of course Sr/r = 0, which corre¬ 
sponds to ^ = 0 . The quadric t = co is the auxiliary (371). 
The two centres of curvature (373) are (q and G being the roots of (377)) 
^^1 = (/"^ + Co = (/-i -f tJr^)2) .... (378); 
and the form of these equations shows that the points are the poles of the tangent 
plane S;y9 = 0 with respect to the two quadrics q and to. 
Ihe equation of the tangent to a line of curvature, r = q xAq may by (375) be 
thrown into tlie form 
-q + yf M/ ^ + t]r^)~^q = y (1 -P y) _ (/“I 4. //i,-i)-ly . (379)^ 
wliere t = or q, and the form of this equation shows that the tangents are the 
generalized normals to the quadrics and to. 
The first form of (379) shows that the tangent t^ touches the quadric to, for 
%(/~^ + q/^“i)-y-i(/-i-f q/i-i)-hy= 0 .... (380), 
as appears on replacing tlie middle function by 
* Because (] = ((/-! ++ 
