SURFACES IN HYPERSPACE. 337 



Hence coinpuring with ^^ ^ lyrsdxrd.Vs , 



yi2 = 0, y-ii = 0. 



Also, comparing ds" with its standard form, 



On = 1 + v~/R", 021 = 1, (I'll =1, a = v-/i?-, 



2h = Sa(")y,.. = 7i2«-i[f'" + i'/W - i"'i"i"], 



$ = Kyiiy22 + y22yii) - yi2y2i = o. 



The dyadic $ vanished identically. Hence hh = l^p- + 55, and H- 

 and 5 must be collinear with h. The indicatrix for a fivisfed curve 

 surface reduces to a line along the vector h, extending from the surface 

 (vertex of degenerate Cone I) to the end of 2h. As $s = 0, the condi- 

 tion G = 0, is satisfied, as must be the case from the reasoning gi\en 

 at the outset. 



By a similar method we may calculate the various quantities 

 arising in the case of any surface expressed in parametric form as 

 y = y(w, v). Let 



dy = radu-]- ndv, m = dy/du, n = dy/dv; 

 ds- = dydy = m-du- + 2m'ndudii + n^dv~ ; 

 On = m-, 012 = ni'ii, 022 = H", 



a — aii022 — 012- = m-n- — (m-n)- = (mxn)" ; 



mxn , ,,, dvcidu + dndv 

 M = — j^dyxdM = — mxnx . 



Let dm. = prfzi + qdv, dn = qdu + rdv, 



p = d-y'dir, q = d'-y/dudr, r = d-y/dv-; 

 ^ = a^^(mxn)«[(mxn)x(p(/u- + 2qdudv + rdv-)]; 

 yn = a-Hinxn)«(inxnxp), yy2 = a-'^(in'xn)-(in.xn-xq), 

 1/22 = o-'(inxn)'(inxnxr). 



