Egan — Linear Complex^ and a certain class of Twisted Curves. 37 



The tetrahedron of reference being that ah-eady chosen, the osculating 

 plane at the point {x) is 



dM {yi - Xi) + fC23 (^2 x, - y, xi) = 0, 



the yi being running coordinates. 

 Hence the equations (2) become 



Oi 



f<3 



as 



Ui 



^U ^23 ^i 



^23 ^^2 — ^14 ^l 



and hence, in the neighbourhood of P, by equations (4), 



ajai = a'tP + higher powers, \ 

 os/ai = b't^ + . . ., ? • 



aj/oi = c'f + . . . / 



(5) 



Now, the equation of the osculating plane can also be written in the form 



2/1 2/2 2/3 Vi 



Uji CC'i Xq Qj^ 



Xi 'X^ x^ x^ 

 X\ Xz x^ x^ 



^AiHi = 0, 



(5a) 



the dots denoting differentiation with respect to t. If we substitute for the *,- 

 and their derivatives the values given by (4), we see at once that A^A^, A,, Ai 

 are of orders q + r - 3, p + r -3, p+ q - S, p + q ■{ r -5 respectively in t ; 

 and hence we can write 



AJAi = a'ti' + higher powers, \ 



A,IA, = h"tP-'- + . . ., j. (6) 



A,/ A, = c"tP-i + . . ., ) 



Now the A; are proportional to the a,-, being the coordinates of the same 

 plane ; hence AijA^ - a,/ai, and the equations (6) are therefore the same as 

 the equations (5), which involves the result stated. 



This second form of the proof may be thus summarized: — The class of a 

 singularity P on a curve, i.e. the number of osculating planes coincident with 

 that at P, which can be drawn through an arbitrary point in the osculating 

 plane at P,i& p - q. The degree of the singularity, i.e. the number of points 

 coincident with P in which the curve is met by an arbitrary plane through P, 

 is r. If the curve belongs to a complex, the degree and the class of the 

 singularity are equal, and hence p - q = r. 



7. The equations (4) represent a cycle with r branches, of degree r and 

 class p-q. The theorem of this section may be stated thus: — The degree of 

 any cycle of the ciorve is equal to its class. 



