169 



VIII. — On Supplementary Curves. 



By Robert Finlay, B.A. of Trin. Coll., Dublin, Professor 

 of Mathematics in Manchester New College. 



[Read March 8<A, 1853.] 



The following paper contains, it is hoped, some contributions 

 to the Theory of Imaginaries in Geometry. It was origi- 

 nally intended to include only such points in that theory as 

 appeared to me to be new; but, having proceeded some length 

 in drawing up the paper, it occurred to me that its utility 

 might be increased by including such an outline of what has 

 already been published on the subject as would make the paper 

 intelligible in itself without perpetual references to other 

 sources of information. Accordingly, the first section con- 

 tains some matter which is not absolutely new ; the principal 

 definitions having been given by M. Poncelet, in his splendid 

 work " On the Projective Properties of Figures" published 

 in 1822, and several of the preliminary theorems having 

 been published in the Mathematician , early in 1846, in some 

 papers of mine " On the Application of Algebra to the Modern 

 Geometry." ' 



The first idea of writing the paper originated in the perusal 

 of a long note at the end of Mr. Salmon's new book " On the 

 Higher Plane Curves." In this note he has combated the 

 two principal theories of imaginaries, as proposed respectively 

 by M. Poncelet, and the late Mr. Gregory, of Cambridge. 

 An able defence of Gregory's theory has just been published 

 in No. XXX. of the Cambridge and Dublin Mathematical 

 Journal. In this paper I have confined myself to Poncelet's 



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