Prof. Cayley on Differential Equations and Umbilici. 379 



formed by the other two planes. The differential equation con- 

 sequently is 



ey( j» 2 — 1) + {« — c)xp = ; 

 or, putting for shortness 



c 

 it is 



y ( j9 2 — 1) -f- 2mwp = 0, 



which is the differential equation previously considered. Hence, 

 writing now h in the place of z, the equation of the curve of cur- 

 vature in the neighbourhood of the umbilicus is 



^==(^+v / a)(^ 2 +2/ 2 +v / n) w " 1 = ]E> +Qv / a, 



where D=mV+«/ 2 ; or, what is the same thing, the equation is 



£ 2 -2PA + P 2 -Q 2 D=0; 



and the equation (in the neighbourhood of the umbilicus) of the 

 curve through the umbilicus is 



so that the umbilicus is a trifid node. In the case however of 

 an ellipsoid or other quadric surface, we have m = l, so that 

 the equation of the curve of curvature in the neighbourhood of 

 the umbilicus is 



h— +#v# 2 +2/ 2 , 



or, what is the same thing, 



h*-2hx-y' 2 =0: 



and for the curve through the umbilicus, in the neighbourhood 

 of the umbilicus, the equation is y 2 =0, so that there is only a 

 single direction of the curve of curvature. The differential 

 equation gives, however, at the umbilicus £>(j» 2 + l)=0; the 

 value p—0 is that which corresponds to the curve of curvature; 

 the other two values p =■ Hh i correspond to the curve (pair of 

 lines) a? 2 -f?/ 2 =0, which is the envelope of the curves of curva- 

 ture, or, more accurately, the envelope of the projections of the 

 curves of curvature on the tangent plane at the umbilicus. 



Blackheath, October 17, 1863. 



2C2 



