PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



example, in the form VI U + Vm V + Vp T — 0. If, in this theorem, we take 

 p = 0, then the original curve is the trizomal Vl U + Vm V + Vn W — 0, T is 



any function = — — (a U + b V+ cTP), where, considering /, m, n as given, a, b, c 



are quantities subject only to the condition — +-r + — = 0, and we have the 



a b c 



theorem of "the variable zomal of a trizomal curve," viz., the equation of the 

 trizomal VlU + Vm~V + Vn W = Q» ma y De expressed by means of any two of 

 the three functions U, V, JV, and of a function T determined as above, for example 

 in the form Vl' U + Vm' V + Vn T — ; whehce also it may be expressed in 

 terms of three new functions 1\ determined as above. This theorem, which occu- 

 pies a prominent position in the whole theory, was suggested to me by Mr Casey's 

 theorem, presently referred to, for the construction of a bicircular quartic as the 

 envelope of a variable circle. 



In the y-zomal curve Vl(Q + Z$) + &c. = 0, if B = be a conic, <j)=0a line, 

 the zomals + L<fr = 0, &c. are conies passing through the same two points 

 9 = 0, $ = 0, and there is no real loss of generality in taking these to be the circular 

 points at infinity — that is, in taking the conies to be circles. Doing this, and using 

 a special notation A = for the equation of a circle having its centre at a given 

 point A, and similarly A = for the equation of an evanescent circle, or say of 

 the point A, we have the y-zomal curve WA° + &c. = 0, and the more special 

 form VIA + &c. = 0. As regards the last-mentioned curve, */lK + &c. = 0, the 

 point A to which the equation A = belongs, is a focus of the curve, viz., in 

 the case v = 3, it is an ordinary focus, and in the case v y 3, it is a special kind 

 of focus, which, if the term were required, might be called a foco-focus ; the 

 Memoir contains an explanation of the general theory of the foci of plane curves. 

 For v — 3, the equation v7A + VmB + VnO = is really equivalent to the 

 apparently more general form VlA° + VmB° + V?iC° = 0. In fact, this last is 

 in general a bicircular quartic, and, in regard to it, the before-mentioned theorem 

 of the variable zomal becomes Mr Casey's theorem, that "the bicircular quartic 

 (and. as a particular case thereof, the circular cubic) is the envelope of a variable 

 circle having its centre on a given conic and cutting at right angles a given 

 circle." This theorem is a sufficient basis for the complete theory of the tri- 

 zomal curve VW + Vm B° + \/rcC~° = 0; and it is thereby very easily seen 

 that the curve v7A° + V'mB" + VnO° — can be represented by an equation 

 \/TR + Vm'B' + v / w 7 C / =0. But for v y 3 this is not so, and the curve 

 VIA + &c. = is only a particular form of the curve VlA° + &c. = ; and the 

 discussion of this general form is scarcely more difficult than that of the special 

 form VIA + &c. = 0, included therein. The investigations in relation to the 

 theory of foci, and in particular to that of the foci of the circular cubic and 

 bicircular quartic, precede in the Memoir the theories of the trizomal curve 



