1877.] Prof. A. Cayley 07i the Bicircular Quartic. 565 



following is a remarkable theorem on this part of the subject. The posi- 

 tive pedal of a bicircular quartic is the inverse of its negative pedal if 

 the centre of one of its circles of inversion be taken as origin. 



Chapter YII. is the last. It contains the theory of reciprocating 

 curves from their tangential and intrinsic equation. Thus, if v=/(0) be 

 the tangential equation, its reciprocal is in polar equations 



^ /(f)sin0 



Again, if s=f((p) be the intrinsic equation of a curve, the polar equation 

 of its reciprocal is 



This chapter contains also the theory of parallel curves. The following 

 is a remarkable property of these curves : — Every focus of any order of 

 the original curve is a focus of the next highest order of the parallel 

 curve. 



The last problem discussed in the paper is the rectification of bicir- 

 cular quartics by elliptic integrals, and the method can be extended to 

 sphero-quartics. This problem, so far as the author is aware, is now 

 solved for the first time. 



The paper is enriched by the addition of a very important annex by 

 Professor Cayley. 



IV. Addition on the Bicircular Qaartic.'^ By A. Cayley, 

 LL.D., P.R.S., Sadlerian Professor of Mathematics in the Uni- 

 versity of Cambridge. Received January 24_, 1877. 



(Abstract.) 



Prof. 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 cer- 

 tain parameters (his 6, 6', 6", 6"', or say) w^, w,, Wg, w 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 in fact differential 

 expressions, AA' and BB' of the form M/?Wp BB' and CC of the form 

 M/?a;„ CO' and DD' of the form M,dio^,DJ)' and A A' of the form IslcJ^, ; 



2 s 2 



