SURFACES IN HYPERSPACE. 337 



Hence comparing with '^ = Xyrsdxrdxs , 



Y,, = v[i"' + r/R'-t"'r"i"] 



yii! = 0, yoo = 0. 



Also, comparing ds~ with its standard form, 



an = 1 + v^/B?, Oei = 1, 0-22 =1, a = v^/R-, 



a(ii) = R^v^ a(i2^ = - /^V^'^ a^^^ = 1 + ^'A', 

 2h = Sa(")yrs = R-v-'[f"' + i'/R^ - f'-ff"], 



* = Kyuy22 + y22yii) - yi2y2i = o. 



The dyadic $ vanished identicalK". Hence hh = |i}JL + 55, and [i 

 and 5 must be coUinear with h. The indicatrix for a twisted curve 

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

 (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 given 

 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 



dj = jndu + ndv, m = dy/du, n = dy/dv; 

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



On = m-, 012 = Hl'Il, 022 = H", 



a = 011O22 — ai2- = m-ii" — (inTi)^ = (inxn)^ ; 



mxn , „, dindu + dndv 



M = — -=— rfyxrfM = — mxnx = . 



Va ^Ja 



Let dm = pdu + <ldv, dn = qdu + idv, 



P = d^y/du-, q = d-y/diidv, r = d'^y/dv~; 

 ^ = a-Kmxn)«[(inxn)x(pf/jr + 2qdudv + idv'^)]; 

 yn = a-i(mxn)'(mxnxp), y^o = o-i(mxn)«(mxnxq), 

 2/22 = a-'^{ra.xn)'(nixnxT). 



