'285 
PKOFESSOK C. J. JOLY ON 
QUxVTEliNIONS AND rROJECTlVE 
GEOMETRY. 
(^3 h)J ^—^ 3(7 ^ ~ ^ (y ^ y . . . . (381); 
and, moreover, the lines of curvature form a conjugate rescan 011 the surface, for 
(380) gives 
Sr Jr, = 0 if =/-i (/-i + tjij-j r, =/-i (/"i + UrJ^q . (582), 
(compare (379)), 
The other usual properties analogous to those for confocals may be easily obtained, 
but it must suffice to state that the centres of curvature for the (piadric are 
q, = {f~^StJ-JJ . (383). 
92. lo leduce the equation (377) to a rpiadratic, let the syndjolic nuartic of 
A-i/be 
{h-jy-WJ]rJf + W'{hr\fy~W{}rJ)-^^ = o . . (384); 
then on multiplying by and dividing by 1 + the result is 
+ t{{lrJ)-W'']~l = -^,{lJrth-J)-K . . . (385). 
Observing that the coefficient of on the left is — N fj or — N f-^h, the 
equation (376) becomes 
+ mq/{(h~\ff - N'" (/i->/) + N"( q 
- tSfq {h-J- N'"] q + Sqfq = 0 . . . . (386) ; 
and this immediately reduces to 
iTOq/^^ + 6S7r(/(,-y/r-i~N''7.-i)p + 87 . 74 - 1 ^ 3^0 . . . (387), 
when we replace yi/ by j), and discard the extraneous factor t. 
If n and are the fourth invariants of/" and /(, N = ; and it is easy to see 
that n is the result of substituting q in the equation of the Hessian of the surface 
if Q is an integral as well as a homogeneous function of q. Thus one root is infinite in 
either of two cases, if the point is on the Hessian, and if it is on the auxiliary quadric ; 
in either case the centre of curvature is the pole of tlie tangent plane with respect 
to the auxiliary. A root is zero if Sph~^p = 0, and in this case the tangent plane 
touches the auxiliary, and a centre of curvature is the point q itself These special 
cases depend on two distinct conditions, the relation of the auxiliary quadric to the 
surface, and the relation of the Hessian to the surface, 
93. A curve is a generalized geodesic when consecutive tangents are coplanar 
with the pole of the tangent plane with respect to the auxiliary quadric; or, 
symbolically, 
