414] 
ON POLYZOMAL CURVES. 
471 
two trizomal curves, each expressible by means of any three of the four functions 
U, V, W, T\ for example, in the form VZ'ZJ + \fm'V + \/p'T = 0. If, in this theorem, 
we take p — 0, then the original curve is the trizomal *JlU + \ZviV + *JnW = 0, T is 
any function = — ^ (a I/ T + b F + cW), where, considering Z, m, n as given, a, b, c are 
quantities subject only to the condition - + ^ + - = 0, and we have the theorem of 
£1 D C 
“ the variable zomal of a trizomal curve,” viz. the equation of the trizomal 
dlÙ + VmV -f \/nW = 0, may be expressed by means of any two of the three functions 
U, V, W, and of a function T determined as above, for example in the form 
VZ'Z7 + Vm'F + *Jn'T = 0 ; whence also it may be expressed in terms of three new 
functions T, determined as above. This theorem, which occupies 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 VZ (0 + A<E>) + &c. = 0, if 0 = 0 be a conic, <I> = 0 a line, 
the zomals 0 + E> = 0, &c. are conics passing through the same two points 0 = 0, 
<I> = 0, and there is no real loss of generality in taking these to be the circular points 
at infinity—that is, in taking the conics to be circles. Doing this, and using a special 
notation 31° = 0 for the equation of a circle having its centre at a given point A, 
and similarly 21 = 0 for the equation of an evanescent circle, or say of the point A, 
we have the z/-zomal curve VZ2P 4- &c. = 0, and the more special form VZ21 + &c. = 0. 
As regards the last-mentioned curve, VZ21 + &c. = 0, the point A to which the equation 
21 = 0 belongs, is a focus of the curve, viz. in the case v — 3, it is an ordinary focus, 
and in the case v > 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 VZ2l + Vm23 + VmQT = 0 is 
really equivalent to the apparently more general form ^Z2T + Vm33° + VViQ£° = 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 trizomal 
curve VZ2t° + Vm23° + \/?i(5 0 = 0 ; and it is thereby very easily seen that the curve 
VZ2l° + + Vn(5 0 = 0 can be represented by an equation \fl'W + Vm'S' -I- vVOP = 0. 
But for v > 3 this is not so, and the curve VZ21 + &c. = 0 is only a particular form of 
the curve VZ2l° + &c. = 0 ; and the discussion of this general form is scarcely more 
difficult than that of the special form VZ21 + &c. = 0, included therein. The investi 
gations 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 s/lA° + Vm33° + Vnd° = 0, and the tetrazomal curve VZ2F + Vm33° -I- VVi(T + VpT)° = 0, 
to which the concluding portions relate. I have accordingly divided the Memoir into 
four parts, viz. these are—Part I., On Polyzomal Curves in general; Part II., Subsidiary