300 WILSON AND MOORE. 



his work.) Now (h'^p.'^S)- is Gibbs's invariant $3 or | $ | for the dyadic 

 4> = hh — |X|x — 55 = |yiiy22 + |y22yii — yi2yi2. We may therefore 

 write as the projective invariant 



<^3 = 1*1 = i(yiixy22xyi2)" = o. 



In case $3 5^ 0, we can find a second standard form for the develop- 

 ment of a surface about a point. It has been shown that if we project 

 a surface on the S3 determined by the tangent plane and a normal 

 parallel to an element of Cone II, the projection has total curvature 

 null. By taking an element of Cone III and two perpendicular ele- 

 ments of Cone II as axes, the expansion to second order terms becomes 



zi = i(^.^■' + 2Bx!, + Cxf), G = AC - B\ 



22 = K^i-t' + 2B,xy + Cif), = JiCi - B,\ 



^ — 1 

 ■^3—2 



{A^^x- + 2B-2XII + C^rf). = .•I2C2 - B.}. 



We shall show that by a proper choice of the element of Cone III, 

 the standard form 



z, = UAx" + 2Bxy + Ci/)., zo = ^Dx^ z, = ^Fa/ (123) 



may be found. All that is necessary to prove this is to prove that the 

 two double lines obtained from zo = and 23 = may be made per- 

 pendicular. If we set ^ = vh, rj = ue -\- vf + wA, f = tvB. The 

 condition A = becomes f ^ + 77^ — ^^ = 0. We have to find two 

 directions u, v, w, such that 



fr + y]i- - ^1- = 0, fs' + 7?2^ - f_2 ^ 0, u,w, + i^v-i + iciico = 0. 



Furthermore the double lines (^ + ?7).i'- + (^ — 77)^" + 2^xy = 

 must be perpendicular for the two series of ^, 17, f . Hence, if p be a 

 factor, 



pfe — 772) = ?i + r?i , pfe + 772) = ^1 — 771 . Pf2 = — fi, 



<'!• ?i$2 - ri'72 + Ts^i + r277i = 0, Tife + fi'72 + ^2^1 — r277l =■ 0, 



or t\h + ^2^1 = 0, ^'1772 — iir]i = 0. 



Let ^i^u = H, , 77,/.t,- = Hi] then the five equations are 



Hi" - Hi- + 1 = 0, Jh' -22^+1-0, H2 + Hi = 0, II, - Hi = {) 



