Minutes of Proceedings. 47 



It will be within the knowledge of many of our Members, that of 

 late years mathematicians have been much occupied with the investi- 

 gation and development of a new Algebra, transcending in its power 

 of generalization the methods of analysis previously known. This 

 new Calculus may be said to have originated with my late distinguished 

 and lamented colleague Dr. Boole. Subsequently, it has been deve- 

 loped by the labours of Sylvester and of Cayley in England, by the 

 writings of Eev. Dr. Salmon in this country, and by several eminent 

 foreign mathematicians. Among the most important investigations by 

 which this new branch of Science has been enriched in its geometrical 

 relations are those recently published by our colleague Dr. Casey. 

 To those investigations so much value is attached by competent mathe- 

 matical authorities, that the Council of this Academy did not hesitate 

 to award to Dr. Casey, in recognition of their merit, a Cunningham 

 Gold Medal. I shall attempt, though very briefly, and I fear imper- 

 fectly, to notice the general nature of those Papers. 



Among Dr. Casey's earlier contributions to this higher modern 

 geometry, is a memoir which was published in volume ix. of the 

 Proceedings of this Academy. This Paper contains a number of new 

 and remarkable theorems, with respect to the Contact of Circles and 

 Spheres, as well as some extensions to Conic Sections. But in a historic 

 notice of Dr. Casey's labours, this Paper is of especial interest, inas- 

 much as it contains the germ of those new analytical methods which 

 he has developed with such extraordinary success in his subsequent 

 works. The principal features of these methods are well illustrated in 

 the great Paper on "Bicircular Quartics" which he laid before this 

 Academy in February, 1867, and which appears in volume xxiv. of our 

 Transactions. 



The curves known as bicircular quartics are a peculiar class, in- 

 cluded under the more general designation of curves of the fourth 

 degree. The properties of these curves had been already, to some ex- 

 tent, investigated before the commencement of Dr. Casey's labours, 

 but it was found that the ordinary Cartesian analysis is not sufficiently 

 tractable to be applied to this particular class of curves with advantage ; 

 and consequently, before Dr. Casey's work appeared, the true geome- 

 trical relations of these curves had been but imperfectly apprehended. 

 By a most elegant analytical conception. Dr. Casey placed the true 

 theory of these curves at once in a proper light. In his new system 

 of co-ordinates, the variables denote circles instead of the straight lines 



