330 
[460 
460. 
NOTE ON ME FEOST’S PAPEE ON THE DIRECTION OF 
LINES OF CUEYATUEE IN THE NEIGHBOUEHOOD OF 
AN UMBILICUS. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. x. (1870), 
pp. 111—113.] 
I remark as follows : 
1. In regard to a quadric surface, it is not, I think, correct to say that the 
generatrices through an umbilic are curves of curvature ; notwithstanding that, as shown 
p. 80, the normals at every point of such generatrix lie in one plane and consequently 
intersect. The way in which these generatrices as g?iasf-curves-of-curvature present them 
selves is as follows : 
The curves of curvature satisfy a certain differential equation, the complete integral 
of which gives these curves as the intersections of the given quadric surface by the 
series of confocal surfaces —— r + - 7 — = 1, h being the constant of integration 
a 1 + h b 2 + h c 2 + h 6 6 
of the differential equation. The singular solution of the differential equation, or envelope 
of the curves of curvature determined as above, gives the umbilicar generatrices. 
2. In regard to a surface in general, I think it must be considered, not that 
there pass through the umbilic three distinct curves, but that the umbilicar curve of 
curvature is a curve having at the umbilic a triple point, or rather a point at which 
there are in general three distinct directions of the curve. The umbilicar curve of 
curvature in fact presents itself as the curve belonging to a certain value of the 
constant of integration h ; in order that the curve of curvature may pass through a 
given point on the surface, h must satisfy a certain quadratic equation, that is for a 
given point of the surface there are two values of h, and therefore two curves of