Iviii 



c here denoting a scalar (or real) quantity, which is indepen- 

 dent of the origin of vectors, and seems to have some title to 

 be called the total tension of the system. 



In mentioning finally some applications of his algebraic 

 method to central surfaces of the second order, the author 

 could not but feel that he spoke in the presence of persons, 

 of whom several were much better acquainted with the gene- 

 ral geometrical properties of those surfaces than he could pre- 

 tend to be. But, while deeply conscious that he had much to 

 learn in this department from his brethren of the Dublin 

 School, as well as from mathematicians elsewhere, he ventured 

 to hope that the novelty and simplicity of the symbolic forms 

 which he was about to submit to their notice might induce 

 some of them to regard the future development of the princi- 

 ples of his method as a task not unworthy of their co-operation. 

 He finds, then, that if a and /3 denote two arbitrary but constant 

 vectors, and if p be a variable vector, the equation of an ellip- 

 soid with three arbitrary, and, in general, unequal axes, re- 

 ferred to the centre as the origin of vectors, may be put under 

 the following form 



(ap 4- pa)2 _ (/3p _ p^)-^ - 1. (21) 



One of its circumscribing cylinders of revolution is denoted 

 by the equation 



- (Pp - p/3)^ = 1 ; (22) 



the plane of the ellipse of contact by 



ap + pa =z 0; (23) 



and the system of the two tangent planes parallel hereto, by 

 (ap + pa)' :::: 1. (24) 



A hyperboloid of one sheet, touching the same cylinder in 

 the same ellipse, is denoted by the equation 



{ap + pay + (/3p - p/3)' = - 1 ; (25) 



its asymptotic cone by 



(«p + Qaf + (/3p - p/3)- = ; i2%) 



