460] 
NOTE ON MR FROST’S PAPER &C. 
331 
42—2 
curvature; but an umbilic is a point for which (as in effect shown, p. 81, for the 
particular case of a quadric surface) the two values of h become equal; that is, there 
is through the umbilic only a singular curve of curvature; but ^ is determined by a 
cubic equation, and the umbilic is thus (as just mentioned) a point at which there are 
in general three distinct directions of the curve. 
3. Some researches on the subject are contained in my paper “On Differential Equations 
and Umbilici,” Phil. Mag., vol. xxvi. (1863), pp. 373—379 and 441—452, [330]. It is 
noticeable that in the integral equations which I have there obtained for the differential 
equations cy (p 2 —l)+(a—c)xp=0, and the more general form (bx + cy)(p 2 — l)+2(fx+gy)=0, 
which belong to the neighbourhood of an umbilic, the curve through the umbilic does 
break up into three distinct curves; and the same is the case with the umbilic on the 
surface xyz — 1 presently referred to. 
4. In the paper “ Memoire sur les surfaces orthogonales,” Liouv., t. xii. (1847), 
pp. 241—254, M. Serret has given two very remarkable cases of three systems of surfaces 
intersecting each other at right angles, and consequently in the curves of curvature of 
the surfaces of each system. It was only on referring to this paper, in connexion with 
that of Mr Frost, that I perceived an obvious enough simplification of M. Serret’s 
formulae, whereby it appears that the curves of curvature on the surface xyz = 1 are 
given as the intersection of this surface with the series of surfaces 
h = (a? + wy~ + w-z 2 f + (x 2 + co 2 y 2 + toz 2 )~, 
where co is an imaginary cube root of unity; the rationalised equation is of the 
twelfth order in (x, y, z), and for the particular value h = 0, reduces itself as is easily 
seen to 0 = (y 2 — z 2 ) 2 (z 2 — x 2 ) 2 (x 2 — y 2 ) 2 . The point x = y = z = 1 is obviously an umbilic 
on the surface xyz = 1, and the corresponding value of h being h = 0, the equation just 
obtained determines the umbilical’ curves of curvature, viz. combining therewith the 
equation xyz = 1 of the surface, we have the three hyperbolic curves 
(y = z, xy 2 = 1), (z = x, yz 2 = 1), (x = y, zx 2 = 1).