182 THE ROYAL SOCIETY OF CANADA 



Let us now consider surfaces (5) of range 2 defined by equations 

 (27). The procedure to be followed is the same as that used above. 

 The conditions to be satisfied in order that the curves q may belong 

 to the complexes (28) are: 



(40) a-\-mz— ny = Tdi, 



^-\- nx— /z=T02, 

 7+ ly — mx — Tdz. 



From these, on deriving with respect to p and inserting the values 

 of Xi, yx, Zi as found from (6), we obtain 



(41) Byimd.'-^ nd/) -0/(w02+ nds) = r^i' + r'^i, 



These equations, as before, are not independent; they impose only 

 two conditions on the functions involved. If we eliminate r between 

 the first two, we find 



(42) {6x62') {lex'-^md^'+nd^'-T') = Q. 



The factor (^1^2') can not vanish, since X(^, q) is unequal to zero. 

 It therefore follows that 



(43) M'+W^2'+«^3'-r' = 0. 

 Employing this value of t in (41), we infer that 



(44) /9, + W^2+«^3+r = 0. 



Hence, from (43) and (44) 



r = T{q), 

 and 



(45) ai5i + a202+ 03^3=1, 

 where ai, a2, az are functions of q alone. 



If we insert the value d given by (26) in (45), there results the 

 equation 



(46) ^ / . . ^ 2 

 dp p+q 



We may regard this as a linear differential equation in p; its 

 integral is 



-|-a2(r2+a30-3=(/>+ç)"(/(g)-; — -— I. 



Thus (aio-i + a2<r2+ 030-3) is a quadratic function of p. 



From (46) and (47) we find, by successive differentiation with 

 respect to p, 

 (48) ttio-i +a2<r2 +030-3 =ip-\-qyf(q)-{p-\-q), 



(47) 



ttiOi -f- 020-2 +030-3 



V I w 1 _ 



oio-i -r 020*2 -r 030^3 



oio^i -r 0202 ~r 0303 



= 2(/>+g)/(g)-l, 

 = 2f(q), 

 ' = 0. 



