220 Proceedings of the Royal Society 
It is right that I should explain the very great extent to which, 
in the composition of the present memoir, I am indebted to Mr 
Casey’s researches. His paper “ On the Equations and Properties 
(1.) of the System of Circles touching three circles in a plane ; 
(2.) of the System of Spheres touching four spheres in space ; (3.) 
of the System of Circles touching three circles on a sphere ; (4.) on 
the System of Conics inscribed in a conic, and touching three in- 
scribed conics in a plane,” was read to the Royal Irish Academy, 
April 9, 1866, and is published in their “ Proceedings.” The fun- 
damental theorem for the equation of the pairs of circles touching 
three given circles was, previous to the publication of the paper, 
mentioned to me by Dr Salmon, and I communicated it to Professor 
Cremona, suggesting to him the problem solved in his letter of 
March 3, 1866, as mentioned in my paper, “ Investigations in 
connection with Casey’s Equation,” “Quart. Math. Journal,” t. viii. 
1867, pp. 334-341, and as also appears, Annex No. IV. of the present 
memoir. 
In connection with this theorem, I communicated to Mr Casey, 
in March or April 1867, the theorem No. 164 of the present 
memoir, that for any three given circles, centres A, B, G , the 
equation BG J A° + GA VB° + AB V C° belongs to a Cartesian. 
Mr Casey, in a letter to me dated 30th April 1867, informed me of 
his own mode of viewing the question as follows : — “The general 
equation of the second order (a, b, c, f g , h) (a, [3, y) 2 , where a, /3, y 
are circles, is a bicircular quartic. If we take the equation 
(a, b, c,f, g , h) (X, /x, v) 2 in tangential co-ordinates (that is, when 
\, g., v are perpendiculars let fall from the centres of a, {3 , y on any 
line), it denotes a conic ; denoting this conic by F , and the circle 
which cuts a, /3, y orthogonally by J, I proved that, if a variable 
circle moves with its centre on F, and if it cuts J orthogonally, 
its envelope will be the bicircular quartic whose equation is that 
written down above ;” and among other consequences, he mentions 
that the foci of F are the double foci of the quartic, and the points 
in which J cuts F single foci of the quartic, and also the theorem 
which I had sent him as to the Cartesian, and he refers to his 
paper on bicircular quartics as then nearly finished. An abstract 
of the paper as read before the Royal Irish Academy, 10th Feb- 
ruary 1867, and published in their “ Proceedings,” pp. 44, 45, con- 
