140 On the Contacts of Lines and Surfaces of the Second Order. 



It may not be imiutercsting, under an euristic point of view, 

 to state that the above theory, which, as Avell in ^vhat it accom- 

 plishes as in what it suggests (the author cannot but feel conscious), 

 constitutes a substantial accession to analytical science, arose out 

 of a theorem which occurred to him as likely to be true, in the 

 act of reviewing for the press his paper " On Certain AdtUtions " 

 in the last November Number of this Magazine, and which he had 

 only then time to throw into a foot-note as a probable conjecture. 



Wishing to subject it to an analytical test, he found it necessary 

 to obtain the condensed forms which serve to characterize the 

 confluent contact of conies. In this way he became aM^are of 

 the great utility of these condensed forms, and of the desideratum 

 to be supplied in obtaining a complete list of them applicable to 

 all varieties of contact. The happy thought then occurred to him 

 of inverting the process which he had applied in the treatment of 

 the contacts of conies, in the November Number of the Cambridge 

 and Dublin Mathematical Journal ; for whereas the nature of the 

 contacts was there assumed and translated into the language of de- 

 terminants, he soon discovered that it v.'as the more easy and secure 

 course to assume the relations of eveiy possible inimitable kind 

 that could exist between the complete and minor determinants 

 corresponding to the characteristics, by aid of these relations to 

 construct the characteristics, and from the characteristics so ob- 

 tained, determine the geometrical character of each resulting 

 species of contact. Thus he has been able to effect the very 

 results stated by himself as desiderata at the close of the paper 

 in this Magazine above referred to. 



Note. — It is proper to remark, that all the condensed forms 

 given in this paper have actually been obtained by the author in 

 the way above pointed out. The limits imposed by the objects 

 to which the Magazine is devoted have restricted him from ex- 

 hibiting the method at full } but any of his readers will be able 

 without difficulty to make it out for himself. 



The process consists in finding U-f-XV by means of solving 

 for each case a problem of position (a kind of chess-board pro- 

 blem) on a square table, containing three places in length and 

 breadth for conies, four places by four for surfaces, and so on 

 (if need be) according to the number of variable letters involved. 

 IJ + XV being thus determined in form, U and V become readiiy 

 cognizable. It is right also to add, that some of the condensed 

 ftn-ms here set forth have been incidentally noticed and employed 

 by previous authors, as Pliicker and Mr. Caylcy. 



The conditions in each case to which the position-problem is 

 subject are iunnediately deducible from the laws which the com- 

 plete determinant, and the successive minor systems of determi- 

 nants of U -\- XV, are required to satisfy. 

 [To be continued.] 



