188 Mr. F. L. Hitchccck on " 



If we take e a unit- vector along this direction, and t? 

 another unit-vector such that eij = v, it is legitimate to write 



y v = vyy + exe + VX 1 ? ' 



the vector part of \yv is equal to the term vyy, a result of the 

 paper referred to above ; Ve%e = by the last paragraph ; 

 whence Yrj-^r) also vanishes and rj gives the other root of the 

 strain-cubic. 



2. To illustrate further these fundamental facts, take Dupin's 

 theorem that " each member of one of three families of 

 orthogonal surfaces cuts each member of each of the other 

 families along its lines of curvature/'' 



Let the unit-normals be v v v and v 2 . Then 



Vvi = V(v. 2 v) — Vv 2 . v — v 2 VV — 2^v 2 . 

 Operate by Sj/j, remembering that 



Si/ Vv = S Vi7v 1 — S v 2 X7v 2 — » 

 we thus have at once 



that is, x v 2 is at right angles to v Y . Put ^v 2 is already known 

 to be at right angles to v, and. is therefore parallel to v 2 . 

 This proves the proposition. 



3. In order to study certain quantities related to the 

 second differential of the vector v we may adopt the nota- 

 tion 



dY\7v = \jfdf), 



and remembering that yy and ~SYJv have the same tensor 

 we may put 



Thus X, jx, and v form a rectangular unit system. Differen- 

 tiation with regard to these three directions may be repre- 

 sented by -p:, ■=—. and ~r respectively. Here X and y, are 

 J dV dm' dn l 



not the same as the e and rj of Art. 1, except in certain 

 cases, of which families of cylinders are among the simplest. 

 The constituents of -^r may be arranged according to the 

 following skeleton : — 



ylr/i = r'X + Q/a +p 

 yjrv=gX+joFfL + n 



