Egan — Linear Complex^ and a certain class of Twisted Curves. 63 



Factors of J for the General Bational F-eurve. 

 54. Consider the determinant of the sixth order 





dt dt' '' dd d9' 



where Yij as before is the result of substituting 6 for t in X,j. The 

 function / is of degree 3 (m - 2) = 6»i - 12 in 9 and also in t. Again j 

 contains the factors D{t) and F{9) where -^(0 = | «;«,*,■ if; |. 



To prove this point, we suppose that ts is a singularity with the 

 characteristic numbers p,q,r. Then changing the tetrahedron of reference 

 and writing i! for t - ts, we find that the orders in t of the initial terms 

 of the Xij are 



2) + q- 1, 2} + r - 1, p - 1, q + r-l=p-l, q - 1, r - 1. 



Let us write JTi, JT^, . . . Z^ for the X,y, arranged according to their 

 infinitesimal ■ order in t, Xi being of the highest order. If we substitute 

 X\ = AJTs + fiXi for X3, \ and fi being constants properly chosen, X\ will 

 be of order p (or higher). That being so, it is clear that the term of lowest 

 degree in t belonging to j will occur in the product of 



Xi Xi, Xe 



I Xi Xs -lo 



Xi X5 Xs 

 by its minor in /. 



Now Xi = AtP-' + A'ti' + ..., X, = £ti-' + ..., X, = Ct'-' + ... 

 Hence the term of lowest degree in the determinant last written is 

 ABG{p -q)(q- r) (r - p) tP*i^'-\ 



The coefficient, it is to be noticed, does not vanish, even when, for instance, 

 r = 1. It follows that / considered as a function of t has a zero of order 



p + q + r-6'^2(p-3), 



no more and no less, at a point 4 at which p > 3. 



Hence y contains the factor U.(t - <s)^^"^ which, as we saw in section iv, is 

 equal to I>{t), disregarding constant factors. In like manner, y contains B(6}. 



Again, j is easily seen to contain {9 - tf. Hence 



j{t,9)^D{t)D{9){9-tfF{t,9). 

 The degree in < of i^ is 



(G/i - 12) - (4» - 12) - 9 = 2w - 9. 



