Prof. Cay ley on Differential Equations and Umbilici, 377 



stant of integration. In fact, for this value, the integral equation 

 becomes 



-y 2m \y 2 + {2m-l)^} m - 1 =0, 



which belongs to a set of 2m + (m— 1) + (m—1) lines coi nciding 

 with the lines y = 0, y=isc\/2m—lj and y=— ix\/2m— 1 

 respectively. The directions at the origin are therefore p = 0, 

 p=±i\/im—l, which are the same values of p as are obtained 

 from the differential equation; viz. since this is satisfied identi- 

 cally at the point in question, proceeding to the derived equation, 

 we have 



p{p*-I) + 2mp = 0; 

 that is, 



p{p* + 2m-l)=0. 



But it is to be observed that these values of p are different from 

 the values given by the equation □ = m 2 # 2 + «/ 2 = 0, which are 

 p= +zm. The reason is that the curve □ =0 being, as was 

 shown, a cuspidal curve on the surface, the equation □ =0 is not 

 a solution of the differential equation. 



If however m=l, then the integral equation gives at the 

 origin no longer three values of p, but only the value p = 0. 

 The differential equation however gives, as in the general case, 

 three values; viz. we have p(p 2 + 1)=0. And the values p= +i 

 obtained from the factor p 2 + 1=0 are precisely the values of p 

 obtained from the equation □ =# 2 + 2/ 2 = 0, which in the case 

 now under consideration belongs to the reduced or proper enve- 

 lope of the surface, and is therefore the singular solution of the 

 differential equation. 



III. 



The two curves of curvature which pass through any given 

 point of a surface are distinct curves, not branches of one inde- 

 composable curve. In fact if P, Q are the two curves of curvature 

 for a point A, then for a point A' on P the two curves of curva- 

 ture will be P, Q' ; and if P, Q were branches of an indecom- 

 posable curve, then P, Q' would also be branches of an indecom- 

 posable curve, and we should have P a branch of two different 

 indecomposable curves, which is of course impossible. In the 

 case of an umbilicus, the two curves P and Q coincide together; 

 or, as we may express it, the curves of curvature through an 

 umbilicus are the duplication of a single, in general indecom- 

 posable curve ; and in general this curve has at the umbilicus a 

 trifid node. I use this expression to denote a point at which 

 there are three distinct tangents, or, more accurately, three 

 distinct directions of the curve : an ordinary triple point is of 



Phil Mag. S. 4. Vol. 26. No. 176. Nov. 1863. 2 C 



