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V. — A Study of Trilinear Co-ordinates : being a Consecutive Series of Seventy - 

 two Propositions in Transversals. By the Rev. Hugh Martin, M.A., Free 

 Greyfriars', Edinburgh. Communicated by Professor Kelland. 



(Read 20th March 1865.) 



Introductory Remarks. 



The following series of theorems is given as an illustration of the modern 

 method of trilinear co-ordinates, having been wrought out after perusal of Mr 

 Ferrar's very lucid and elegant treatise on that subject. The demonstrations 

 present no difficulty, requiring nothing more complicated than the formation of 

 determinants of two and three places. Accordingly, after exhibiting the method 

 of proof in a few instances, I have merely given the enunciations of the remaining 

 propositions. As the series of theorems advances the manipulation becomes, of 

 course, a little more complicated ; but the co-ordinates and co-efficients always 

 appear in such symmetry as very greatly abbreviates the task, and guarantees 

 its accuracy. Two, or perhaps three, of these seventy-two theorems are known 

 mathematical truths ; but that so many new consecutive propositions should be 

 so easily found, and so easily proved, is a convincing evidence of the simplicity, 

 fertility, and power of this new and beautiful method. 



Treated according to the ancient geometry, the contents of the following pages 

 would constitute a volume of no mean dimensions ; and some of the propositions, 

 such as those which affirm that the six points P lf P 2 , P 3 , P 4 , P., P G range in a 

 straight line, and that the seven straight lines U^; R X R 2 , R 3 R 4 ; S^, S 3 S 4 ; 

 QiQ 3 , Q-iQ^i Q 2 Q 6 a ^ meet m a P omt 5 would probably have been undiscoverable. 



In the admirable treatises of Mulcahy and Townsend a few analogous propo- 

 sitions are demonstrated geometrically. Mr Townsend, in particular, has a 

 chapter in his first volume, on concurrent lines and co-linear points, which falls 

 in very closely with the kind of propositions which the following series embraces. 

 His second volume I have not been fortunate enough to see ; but the subject is 

 only ripening for a systematic gathering-up of the propositions that have been 

 discovered in this line of investigation, and the following pages are presented as 

 a humble contribution towards that desirable result. 



A word or two may be permitted in reference to the additions to the termin- 

 ology which must be made, and generally sanctioned by mathematicians, ere such 

 systematic digest can be successfully accomplished. I have ventured — of course 

 only provisionally — on one or two such additions. When the co-ordinates of two 

 points are respectively the algebraical inverses of each other, I have called these 

 points, in reference to each other, " inverse points ;" and it is evident that a very 



VOL. XXIV. PART I. L 



