386 Proceedings of the Royal Irish Academy. 



that is, the sphere contains the director circle of the conic as a great 

 circle. A similar result holds for the hyperbola 



p-a cosh u + p smh w = — -— , ii t = tanh f w. 



Taking the equation of the general unicursal cuitc of even order in 

 the standard form given in Art. 9, the invariant of the present article, 

 being the (12)" invariant of the quantic 



pfi^y) - «^ {^y) - P^{^y) - y ^ («y). 



reduces at once to 



p2 - a^ - /32 - y- = 0, 



and the sphere is the director sphere of the quadric 



ISp6-'p = -\. 



Referring to the list of vanishing invariants which is given at the close 

 of the aiticle cited, there is no difficulty in proving this. 



12. Formation of a system of curves called '■'■ Emanants^'' projective with 

 the original curve. 



From any binary quantic n system of emanants may be derived by 

 the aid of operators of the type 



d d 



In connexion with a curve 



' dx dy 



^ 7(^y) 



^{xy) 



X^y) 



of order «, may be considered the emanant curve 



of order p, if x, y are regarded as variable, and Xi, i/i as given. Now, 

 if \p is any linear vector function, the original curve is projected by 

 operating by i/^, and replacing xpp by p. Thus 



, ^ixy) 



