86 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



(A v D x ) be the antipoints of (A,D) and (B v C r ) the antipoints of (B, C) ; so that 

 (A^B V tfp D r ) are another set of concyclic foci. We have Bj . C x = B . C, and it 

 appears, ante No. 104, that we can find l xi m v n v such that identically 



- /A + r/iB + nC = - l x \+ //^BjH- ^C, 



and that m x n x = m n. The equation of the curve gives 



- £A + mB + nC + 2 JmnBC — , 



we have therefore 



— / x Aj + B4BJ+ »jO+ 2 %/mjWjBjC! = , 



that is, 



V/A + Vm^B7+ n//jC 1 =0, 



viz., this is the equation of the curve expressed in terms of the concyclic foci 

 A v x>j, Cj. 



27ic Tetrazomal Curve, Decomposabli or I n decomposable. — Art. No. 194. 



194. I consider the tetrazomal curve 



JlK° + JmB° + J^C 5 + Jp~0° = , 



where the zomals are circles described about any given points A. B, C, J) as 

 centres. 



There is not, in general, any identical equation aA°+ bB + cC° + dD = 0, but 



when such relation exists, and when we have also- + — + — + ^- = 0, then the 



a li c a 



curve breaks up into two trizomals. When the conditions in question do not 

 subsist, the curve is indecomposable. But there may exist between /, m, n,p re- 

 lations in virtue of which a branch or branches ideally contain (z° = 0) the line 

 infinity a certain number of times, and which thus cause a depression in the order 

 of the curve. The several cases are as follows : — 



Cases of the Indecomposable Curve. — Art. No. 195. 



195. I. The general case; /, m, n,p not subjected to any condition. The curve 

 is here of the order = 8 ; it has a quadruple point at each of the points /, J (and 

 there is consequently no other point at infinity) ; it is touched four times by each 

 of the circles A, B, C,D; and it has six nodes, viz., these are the intersections of 

 the pairs of circles 



JvW + JnC° = 0, JTK° + J^D a = , 

 Jn~& + J IK = 0, JmW + Jp~U r = , 

 JW + JmW=0, J n ~C r + 7^=0; 



the number of dps. is 6 + 2 . 6, = 18, and there are no cusps, hence the class is 

 = 20, and the deficiency is = 3. 



