378 Prof. Cayley on Differential Equations and Umbilici, 



necessity a trifid node, but not conversely. The umbilicus of an 

 ellipsoid or otber quadric surface is a peculiar exceptional case. 

 In support of the foregoing conclusions, consider a surface 

 having an umbilicus at the origin, and take z = as the equa- 

 tion of the tangent plane at that point ; the equation of the sur- 

 face in the neighbourhood of the umbilicus will be 



z = | k(x* + y q ) -|- 1 {ax s + Ux^y + Sexy* + dy B ) ; 



so that, writing as usual p and q for the first, and r> s, t for th§ 

 second differential coefficients of z\ we have 



p = kx + J (ax* + 2bxy + cy 2 ),. 



q=fy + i (bx 2 -f 2cxy + dy*)> 



r=k-\-ax + by, 

 s= bx + cy,. 

 t = k + cx + dy. 



The differential equation of the curves of curvature projected 

 on the plane of xy is 



(|) 9 [(1 + ?)s-nt\ + J [(1 + ftr- (1 + P *)(] - [(1 +p*) S -pqr-} =1 



and substituting therein the foregoing values of p, q, r, s, t, but 

 attending only to the terms of the lowest order in (x, y) 3 and 



using moreover in the sequel p in the place of — , the equation 



becomes 



{bx + cy){p' 2 -l) + [{a-c)x+(b-d)y']p=:0', 



which may be taken as the differential equation of the curves of 

 curvature at and in the neighbourhood of the umbilicus. The 

 equation is satisfied identically by the values x=0, y = Q f which 

 correspond to the umbilicus ; and to find p, we have to differen- 

 tiate the equation, and then substitute these values of % and y ; 

 we thus obtain 



(b + cp){pZ-l)+[(a-c) + {b-d)p]p=0 > 



or, what is the same thing, 



p{a + 2bp + cp % )-{b-j-2cp + dp*) = 0» 



a cubic equation for the determination of p. 



I remark that we may without loss of generality write d=0) 

 but to simplify the investigation, I suppose in the first instance 

 that we have also b = 0. This comes to assuming that one of 

 the three planes ax 3 + Zbx' 2 y + Zcxy 2 + dy 3 =:0 bisects the angle 



