667] 
223 
667. 
ON THE BICIRCULAR QUARTIC: ADDITION TO PROFESSOR 
CASEY’S MEMOIR “ON A NEW FORM OF TANGENTIAL 
EQUATION." 
[From the Philosophical Transactions of the Royal Society of London, vol. clxvii. 
Part II. (1877), pp. 441—460. Received January 24,—Read February 22, 1877.] 
Professor Casey communicated to me the MS. of the Memoir referred to, and he 
has permitted me to make to it the present Addition, containing further developments 
on the theory of the bicircular quartic. 
Starting from his theory of the fourfold generation of the curve, Prof. Casey 
shows that there exist series of inscribed quadrilaterals ABCD whereof the sides AB, 
BC, CD, DA pass through the centres of the four circles of inversion respectively; 
or (as it is convenient to express it) the pairs of points {A, B), (B, C), (C, D), (D, A) 
belong to the four modes of generation respectively, and may be regarded as depending 
upon certain parameters (his 6, 6', 6", 6'", or say) w l , w. 2 , co 3 , co respectively, any 
three of these being in fact functions of the fourth. Considering a given quadrilateral 
ABCD, and giving to it an infinitesimal variation, we have four infinitesimal arcs 
A A', BB', CC', jCD'; these are differential expressions, A A' and BB' being of the form 
Jf,dw 1 , BB' and CC' of the form M. 2 d(o 2 , CC' and DD' of the form M..dw 3 , DD' and 
AA' of the form Mda>; or, what is the same thing, AA' is expressible in the two 
forms Mdci) and M 1 dco 1 , BB' in the two forms M 1 dw 1 and M. 2 dco. 2 , &c., the identity of 
the two expressions for the same arc of course depending on the relation between 
the two parameters. But any such monomial expression Mdoo of an arc A A' would 
be of a complicated form, not obviously reducible to elliptic functions; Casey does 
not obtain these monomial expressions at all, but he finds geometrically monomial 
expressions for the differences and sum BB' — AA', CC' — BB', DD' + CC', DD' — AA' 
(they cannot be all of them differences), and thence a quadrinomial expression 
A A' = A r j dw 1 + N. 2 do), + N s doo s + Ndw (his ds' = p dd + p dd’ + p" dd" + p" dd"'); and that 
without any explicit consideration of the relations which connect the parameters.