360 WILSON AND MOORE. 



his work.) Now (hx|i.^5)2 is Gibbs's invariant $3 or ] $ | for the dyadic 

 <S> = hh — |J.|J. — 65 = ^yiiy22 + |y22yii — yi2yi2. We may therefore 

 write as the projective invariant 



$3=1^1 = Kyii^y22xyi2)^ = 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 = l{Ax^ + 2Bxy + Cy''), G = AC - B\ 

 Z2 = hUix" + 2Bixy + Ci2/2), = ^iCi - B^\ 

 23 = \{A.x'' 4- 2B2xy + (72^2)^ q = A2C2 - B^^ 



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

 the standard form 



z, = ^(Ax^ + 2Bxy + Cy^), 22 = iDx\ 23 = hEy"" (123) 



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

 two double lines obtained from Z2 = and 23 = may be made per- 

 pendicular. If we set ^ = uh, r] = ue -{- vf -\- wA, f = toB. The 

 condition A = becomes ^^ -\- rj^ — ^ = 0. We have to find two 

 directions u, v, w, such that 



ri' + ^1' - ?l' = 0, fa' + 772^ - a' = 0, UiU. + V,V2 + IVlWo = 0. 



Furthermore the double lines (^ + r])x'^ + (^ — v)!/^ + 2fa:?/ = 

 must be perpendicular for the two series of ^, 77, f . Hence, if p be a 

 factor, 



p(?2 — V2) = ^1 + vi, pfe + Vi) = ^i — m, pf2 = — Ti > 



or ri?2 — f 1172 + ^2^1 + f27?l = 0, f 1^2 + ^1772 + ^2^1 " r2'7l = 0, 



or fl^2 + ^2^1 = 0, fl772 — ^21?! = 0. 



Let ^i/ti = Ht, Vi/^i = I^i'j then the five equations are 



Hi" - Si2 + 1 = 0, Hi' - S22 + 1 = 0, S2 + Si = 0, H, -Hi = 



