180 THE ROYAL SOCIETY OF CANADA 



Finally, if we insert the value of 6 as given by (26) in the equations 

 (6), we obtain the following formulae which (with (13)) are those we 

 had in view: 

 (27) 2 



(o-2(r3') + 



y= — (o-30-i') + 



P+q 



Z= — (0-i0-2') + 



p-\-q 



{(T2<rz") dp, 

 {<xz'<y,")dp, 

 WW) dp. 



Applications. 



In this section of the paper it is proposed to discuss the restrictions 

 to be imposed upon the otherwise arbitrary functions appearing in 

 the equations of (5) in order that the twisted asymptotic curves on 

 (5) may belong to linear complexes. 



We shall first consider surfaces of range 1, i.e., those corresponding 

 to 



X = 0. 



Let the one parameter family of linear complexes which contain 

 the curves q be defined by the equation 



(28) i:l(y8z-zÔy)-\-Za8x = 0, 



where 2 denotes the cyclic sum over the three axes, and a, /3, y, I, m, n 

 are functions of q alone. Then on identifying the tangent plane to 

 (S) at the point {x, y, z) with the polar plane of this point in the com- 

 plex (28), we are led to the following relations: 



(29) a-\-mz — ny_^-\-r]X — lz_'Y-\-ly — mx_ 



di 02 03 



If we differentiate these relations with respect to p and take 

 account of (10), we find 



(30) mzi- nyi = T'di-^Ttii', 



nxi — Izi = t'02-\-tix2, 



lyi—mxi = t'0z-{-thz. 



The functions Xi, yx, Si can be obtained at once from (13) ; on inserting 

 them in (30), we obtain 



(31) T'0i + rM/ = ^i(wM2'+ nixz')-{mB2+ ndz)jjn', 



T'^2+rM2' = 02(«At3'+ W)-(n03-{- /0i)m2', 

 T%-\-TfXs' = es( /mi' + WM2')-( I0l + md2)fl3. 



These equations are not all independent; any one is a consequence 

 of the remaining two. If we eliminate t between the first two, we find 



(32) (r'-W-mfii'-nnz') (0m') = 0. 



