lviii 
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 9 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 + pa)? — (Be — of)” = 1. (21) 
One of its circumscribing cylinders of revolution is denoted 
by the equation 
— (Be — eB = 1; (22) 
the plane of the ellipse of contact by 
ap + pa = O; (23) 
and the system of the two tangent planes parallel hereto, by 
(ag + ea)’ = 1. (24) 
A hyperboloid of one sheet, touching the same cylinder in 
the same ellipse, is denoted by the equation 
(ap + ea)” + (Be — eB)’ = — 13 (25) 
its asymptotic cone by 
(ap + ga)’ + (Be — 98) = 05 (26) 
OO a 
