48 Royal Irish Academy. 



to which we are accustomed in the Cartesian co-ordinates ; and the 

 general equation of the second degree, with this new interpretation of 

 the variables, denotes a hicircular quartic in its most general form. 

 The advantages of this mode of representation are easily seen. In the 

 first place we are at once enabled to utilize our abundant information 

 about the properties of a binary quartic in order to ascertain the pro- 

 perties of a bicircular quartic, while in the second place the peculiar 

 analytical mechanism is exquisitely adapted to the investigation of the 

 focal properties which are of so much importance in curves of thi& 

 description. 



The powerful instruments of research thus created by Dr. Casey 

 have been applied by him to the study of the properties of these curves, 

 and he has not only discovered a large number of new theorems, but 

 he had so co-ordinated and arranged the whole theory as to constitute 

 the bicircular quartics into an important branch of modem Geometry, 



The conspicuous success of this analytical method in the plane has 

 naturally suggested to Dr. Casey the study of the corresponding theory 

 in space, of three dimensions. Little was indeed known of the im- 

 portant class of surfaces called Cyclides, until Dr. Casey brought his 

 new analysis to bear upon them. As one of the most immediate con- 

 sequences of this application, he discovered that a Cyclide is the enve- 

 lope of a variable sphere, whose centre moves in a given quadric, and 

 which cuts a given fixed sphere orthogonally. The complete develop- 

 ment of the theory of the Cyclide was published in The Philosophical 

 Transactions for 1871. 



In his next great memoir, " On a j^ew Porm of Tangential Equa- 

 tion" {Phil. Trans., 1877), Professor Casey has turned his attention to 

 a somewhat different department of mathematical research. The co- 

 ordinate of a variable line may be defined by the angle which it makes 

 with a fixed line, and by the length of the intercept measured from a 

 fixed point on the fixed line. An equation between these co-ordinates 

 is termed by Dr. Casey the tangential equation of the envelope of the 

 line. This form of equation lends itself with surprising facility to many 

 geometrical investigations, of which abundant examples will be found 

 in the memoir referred to. One of these discoveries is so important 

 as bearing on Dr. Casey's previous work, that it must not be passed 

 over without remark : I allude to the rectification of the bicircular 

 nartic by the aid of elliptic functions. 



Abundant references in the writings of the most eminent mathe- 



