Egan — Linear Complex, and a certain class of Twisted Curves. 49 

 Let each of these fracbioDS be equal to 1/A. Then 



with sunilar expressions for az, 03. Hence 



&,2 ^2 + &13 «3 + &14 «4 Sal Xl + &23 «3 + ^21 *'4 







&,2 



&13 



Ju 



X 



.-»! 



;?2 



X-i 



Xi 



&=: 







&23 



&24 



ii 



^2 



X3 



Xi 



XM,2 = 



= &12 2&y X,-,- - Zu (612634 + 631 624 + 623 6,4) 

 = - Xu (613 631 + 631 J24 + 623 614). 



But ^i2/-r34 = jii = const. ; hence X is a constant. We may clearly take 

 this constant as unity. The functions a.j of equations (15) will then be equal 

 to the iij : hence the equations (17) are satisfied, B [yy) s ; and hence 

 (33) = 0. 



Hence the necessary and sufficiervt condition that an algebraic P-ctirve should 

 belong to a linear complex is (33) = 0, the functions [x] and (a) being chosen 

 as in sections iv and v. We may note that when the functions are so chosen 

 in the case of a complex curve, the a; will be equal and not merely proportional 

 to linear functions of the Xt, and conversely. 



31. Corollary 1. — In order that any algebraic P-cnrve should belong to a 

 linear complex, it is necessary and sufficient that the function 



W, {t, 6) = 2a.- id) Xi (0 + 2a,- (0 Xi (0) (18) 



should vanish for all values of t and 9. 



First, it is sufficient ; for if we differentiate W^ = three times with 

 respect to 9 and three times with respect to t, and then put 6 = t, we get 



(33) = 0. 



Secondly, it is necessary. For if the curve belongs to a linear complex, 

 we have seen that 



Oj = ^OrjjXj, aij = const., «„ + a^,- = «,-,- = 0. 

 Now, put 



a<(^) = i3,-, x,{9) = y, a,(0 = ".-. ^,-(0 = '-,- 



Then if we substitute for at and /3, m terms of the xt and yi in Wi, we find 



a series of terms of the form 



(ciij + aj,) {xfljj + xjy,) 



which all vanish. W^ therefore vanishes identically for any curve of a linear 

 complex. 



B.I.A. PROC, VOL. XXIX., SECT, A. [7J 



