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XII. On the Bicircular Quartic. — Addition to Professor Casey’s Memoir “ On a new 
Form of Tangential Equation.” By A. Cayley, LL.D., F.B.S., Sadlerian Pro- 
fessor of Pure Mathematics in the University of Cambridge. 
Received January 24, — Read February 22, 1877. 
Professor Casey communicated to me the MS. of the foregoing Memoir, 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 b, O', b", b 1 ", or say) aj y , a 2 , a 3 , a 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 AA', BB', CC', DD'; 
these are differential expressions, A A' and BB' of the form M x da u BB' and CC' of the 
form M 2 d&> 2 , CC' and DD' of the form M 3 du 3 , DD' and AA' of the form M da ; or, what 
is the same thing, AA! is expressible in the two forms ALda and BB' in the two 
forms MjC&y and M 2 du 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 M du of an arc AA' 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' — A A', CC' — BB', 
DD'-j-CC', DD'— AA' (they cannot be all of them differences), and thence a quadrinomial 
expression (his ds'=^gdb-\-g'db' -\-fdb" -\-f'db ") ; and 
that without any explicit consideration of the relations which connect the parameters. 
I propose to complete the analytical theory by establishing the monomial equations 
AA'=M^=M 1 (Zs; 1 , &c., and the relations between the parameters w, u v , a. 2 , co 3 which 
belong to an inscribed quadrilateral ABCD, so as to show what the process really is by 
which we pass from the monomial form to a quadrinomial form 
AA' (or 6?S)=N(&y -j-N 2 c ?& , 2 ~h 
wherein each term is separately expressible as the differential of an elliptic integral ; 
and to further 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. 
3 R 
MDCCCLXXVII. 
