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ON THE EQUATION OF THE TANGENT AT A GIVEN POINT 

 ON A UNI-NODAL QUARTIC CURVE. 



By REV. W. R V7. ROBERTS, D.D., S.F.T.C.D. 



Read Juki 2l. 1918. Published Jamvarx 31, 1919. 



The carve I propose to discuss in this paper belongs to a class of curves 

 which 1 ventured to call bicursal in a former communication made to this 



Academy. We Buppoae that the c rdinates of any curve of this class are 



expressed in terms of a parameter, and in Buch a manner, that to a given 

 value cif the parami spond two j-'ints on the curve, which points 



_ points. The curve in question 1 have called the 

 nni-nodal quartic, the equation <>f which, when the axes of < and y pass 

 through the d written in the form 



J. _ C-0, 



where A J: an binary quantics of the second, third, and fourth 



grees in x and y It is easy to see how, by a proper choice 



of I tion of the axis of :. the equation of the curve can be reduced 



to the form 



A2- ■!:. AQ -0, (1) 



where Q is a binary quantic of the second degree in x and y. Since this 

 ictdon can l>e effected in < >ne way only, it follows that the curve is 

 completely determined when the axi- ind the three binary quantics 



.1. 7. and Z." _ en; and consequently any covariant of the curve can 



be expressed in terms of : and these quantities and their invariants and 

 co variants. 



If we call X, }'. and Z the coordinates of any point of the curve, it is easy 

 to see that we may write 



X = A 



Z JJ - v 1> 



