224 
ON THE BICIRCULAR QUARTIC. 
[667 
I propose to complete the analytical theory by establishing the monomial equations 
A A' = Mdco = M l d(t) 1 , &c., and the relations between the parameters co, cox, co 2 , co s which 
belong to an inscribed quadrilateral A BCD, so as to show what the process really is 
by which we pass from the monomial form to a quadrinomial form 
AA (or dS) = Ndco -(- Nxdwx -H N 2 dco 2 q- N 3 dco 3 , 
wherein each term is separately expressible as the differential of an elliptic integral; 
and further to develop the theory of the transformation to elliptic integrals. We 
require to establish for these purposes the fundamental formulae in the theory of the 
bicircular quartic. 
I remark that in the various formulae f, g, 6, 0 X , 0 2 , 0 3 are constants which enter 
only in the combinations f+ d, f— g, 0 X — 6, 0 2 — 0, 0 3 — 0: that X, Y are taken as 
current coordinates, and these letters, or the same letters with suffixes, are taken as 
coordinates of a point or points on the bicircular quartic: and that the letters (x, y), 
( x \, yd), (%2, yd), yd) are used throughout as variable parameters, viz. we have 
(f+0)x> +(g + 0)y 2 =1,; 
(f+ 0d) Xx 2 + (g + 0d) yd = 1, 
(/+ 0d) x? + (g + 0 2 ) yd = 1, 
(/+ 0 S ) + (g + 0 a ) yd = 1; 
so that x, y = > J^ 1P , are functions of a single parameter co, and similarly 
(xx, yd), {oc2, yd), (oc->, yd) are functions of the parameters &) 1; co 2 , co 3 respectively. We 
sometimes use these or similar expressions of (x, y), &c., as trigonometrical functions 
of a single parameter; but we more frequently retain the pair of quantities, considered 
as connected by an equation as above and so as equivalent to a single variable 
parameter. 
Formulae for the fourfold generation of the Bicircular Quartic. Art. Nos. 1 to 5. 
1. We have four systems of a dirigent conic and circle of inversion, each giving 
rise to the same bicircular quartic: viz. the bicircular quartic is the envelope of a 
generating circle, having its centre on a dirigent conic, and cutting at right angles 
the corresponding circle of inversion; or, what is the same thing, it is the locus of 
the extremities of a chord of the generating circle, which chord passes through the 
centre of the circle of inversion, and cuts at right angles the tangent (at the centre 
of the generating circle) to the dirigent conic; the two extremities of the chord are 
thus inverse points in regard to the circle of inversion. The four systems are 
represented by letters without suffixes, or with the suffixes 1, 2, 3 respectively; and 
we say that the system, or mode of generation, is 0, 1, 2, or 3 accordingly. 
2. The dirigent conics are confocal, and their squared semiaxes may therefore be 
represented by f+0, g + 0: f+0x, g + 01- f+02, d + f+03, 9 + 03, (which are, in