66 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



where the focus r = 0, r'= is coney clic with the other two foci; but from the 

 equation of the curve V = v 'pv f qq, that is we have 



qq 1 + 2X Jpp'qq' + ^ 2 pp' = Krr' , 



or, what is the same thing, 



viz., this is a form of the equation of the curve ; substituting for p,p', q, </, r, r 



their values, writing also 



A = rg - az) (n - a'z) , 



B = (| - 8z) („ - *) , 

 C = (5 - 7 : ) (« - 7 Z ) > 



and changing the constants X, iT (viz. \\\:K = Jl: Jm : */w) the equation is 



V/A + s/mB~+ V/TC = 0, 



viz., we have the theorem that for a bicircular quartic if (£ — az = 0, tj — a'2 = 0), 

 (£ — /fe = 0, »? — j8'z = 0, (£ — 7s = 0), »/ — 7's = 0) be any three concyclic foci, 

 then the equation is as just mentioned ; that is, the curve is a trizomal curve, the 

 zomals being the three given foci regarded as 0- circles. The same theorem holds 

 in regard to the circular cubic, and a similar demonstration would apply to 

 this case. 



158. It may be noticed that we might, without proving as above that the 

 two foci (p = 0, p' — 0), (q — 0, <f — 0) were concyclic, have passed at once 

 from the form pp'qq = V 2 , to the form y/Xpq/ + *Jqrf + Kjrr' = (or 

 JlK = *JmB = »/nC = 0), and then by the application of the theorem of the 

 variable zomal (thereby establishing the existence of a fourth focus concyclic with 

 the three) have shown that the original two foci were concyclic. But it seemed 

 the more orderly course to effect the demonstration without the aid furnished by 

 the reduction of the equation to the trizomal form. 



Part IV. (Nos. 159 to 206). — On Trizomal and Tetrazomal Curves where the Zomals 



are Circles. 



Tlic Trizomal Curve — The Tangents at I, J, Sc. — Art. Nos. 159 to 165. 



159. I consider the trizomal 



*/W + *Jm~B° + JnC = , 



where A, B, C being the centres of three given circles, A , &c. denote as before, 

 viz., in rectangular and in circular co-ordinates respectively, we have 



A = (x - az) 2 + (y - az) 2 - a" 2 z* , = (f - az) (» - a'z) - d^z 2 , 

 B° = (x - bzf + (y - b'zf - b" 2 z 2 , = (g - 03) (v - 0z) - b" 2 z 2 , 

 C° = (x - czf + (y- c'zf - c" 2 z 2 , = (f - yz) (j, - y'z) - c" 2 z 2 . 



