PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



41 



all positions of the point P; taking P to be the other focus, its squared distances 

 are (k — *jAf, &c, whence for the first-mentioned focus we have 



aA + bB + cC + dD = a(£ - ^A) 2 + b(* - N /B) 2 + c(fc - Jc? + d(ft - Jd) 2 ; 



or recollecting that a + b + c + d = 0, we have for the locus in question 

 a^/A^+ b^/B + c^/C + d^/D = ; this locus will be discussed in the sequel. 

 I remark here, that in the case where the four points are on a circle, then (as 

 mentioned above), the axes of the several conies are in the same fixed directions ; 

 there are thus two sets of foci, those on the axis in one direction, and those on 

 the axis in the other direction ; it might therefore be anticipated, and it will 

 appear, that in this case the tetrazomal breaks up into two trizomal curves. 



Formulae as to the two Sets (A, B, C, D), and (A v B v C v Z^), each of four Concyclic 



Points— Art. Nos. 93 to 98. 



93. Consider the four points A, B, C, D on a circle, then taking, as before, their 

 circular co-ordinates to be (a, a', 1), (6, /3', 1), (7, <y', 1), («5, S\ 1), the condition that 

 the points may be on a circle is 



= o, 



1, a, a', 



aa 



1, ft ft, 



/3ft 



!j 7> 7> 



77 



1, 8, b\ 



as' 



viz., this equation may be written 



- 7) («-*): (7 - «) - 8) : (• - 0) (7 - 5) 

 = (/3 - 7 ') («'-*') : (/-«') (ft- 5') : (a'-ft) (y'-8') ; 



or, if for shortness, we take 



6 = y- «, g = P-&, V = </-*!, g =$-&', 

 c = a — j8 , h= y — d, c' =■ a — ft , h' = y — d' , 



and consequently 



a/ + 6gr + c7t = , a/ + &y + e'h' = , 



a = g — h , a' = g — h' , 



b = h-f, b'=h'-f, 



e = f~g, c'=f'-g', 



a + b + c = , a' + &' + c' = 0, 



then the equation is 



af:bg: ch = af ; Vg : c'h' . 



94. Let a,b,c,d, denote as before (a : b : c : d = BCD : — CD A : DAB : —ABC), 

 then we have 



a : b : c : d = 



VOL. XXV. PART I. 



ft ft, 1 



: — 



7> 7> 1 





8, 8', 1 



: — 



a, a', 1 



7> 7> 1 





8, 8', 1 





a, a, 1 





ft ft, 1 



8, 8', 1 





a, a, 1 





ft ft, 1 





7, 7, * 



