1894.] Solutions of Simultaneous Differential Equations, $c. 277 



a reciprocal curve are then obtained, but the line of proof cannot be 

 included in this abstract. 



In the latter part of the paper, the differential equations of the 

 quadriquadric curve are integrated to obtain the absolute invariant 

 of the curve, and the conditions to be satisfied by the four excubo- 

 quartics having closest contact with a curve at a point are found. 



The equations to a quadriqnadric curve referred to the canonical 

 axes at a point of the curve, are 



u = yx?p 3 (xzy z ) = 0, 

 v = 2 xy- 

 where pi = 6 , PZ = 7/6 5 2?s = /3 7 /e. 



'p 3 (xzy z ) =0, "1 



r jpjz 2 p 2 (^^ 2/ 2 ) = 0,J 



These represent two out of the family of quadrics which contain 

 the quadriquadric ; the quadric represented by v = is that which 

 touches at the origin the osculating plane to the quadriquadric at the 

 origin. If we call the fourth point, at which that osculating plane 

 meets the curve, the tangential of the origin, the quadric repre- 

 sented by u = is that quadric of the family which touches, at the 

 tangential, the osculating plane at the tangential. 



These two quadrics are called the canonical quadrics at the point, 

 and it is shown how to find the equations to the canonical quadrics at 

 any point and how to express the canonical invariants at any point in 

 terms of those at the origin. The relations between the invariants at 

 a point and its tangential are found, and lead to the discussion of the 

 singular points indicated by p^ p z p 3 = and^ 2 pip 2 ps p* = 0. 



XL " On the Singular Solutions of Simultaneous Ordinary 

 Differential Equations and the Theory of Congruences." 

 By A. C. DIXON, M.A., Fellow of Trinity College, Cam- 

 bridge, Professor of Mathematics in Queen's College, 

 Galway. Communicated by J. W. L. GLAISHER, Sc.D., 

 F.R.S. Received June 7, 1894. 

 (Abstract.) 



1. This paper is an attempt to shew how the singular solutions 

 of simultaneous ordinary differential equations are to be found either 

 from a complete primitive or from the differential equations. 



The latter question has been treated by Mayer (' Math. Ann.,' 

 vol. 22, p. 368) in a somewhat different way, but with the same 

 result. He also gives a reference to a paper in Polish by Zajaczkow- 

 ski (summarised in vol. 9 of the 'Jahrbuch der Fortschritte der 

 Mathematik '), and to one by Serret in vol. 18 of ' Liouville's Journal.' 



