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of Edinburgh, Session 1867 - 68 . 
cent circle, or say of the point A, we have the v-zomal curve 
VIA 1 + &c. = 0, and the more special form VIA + &c. = 0. As 
regards the last mentioned curve, VIA + &c. = 0, the point A, to 
which the equation A = 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 equa- 
tion VIA + VmB + V n C — 0 is really equivalent to the appa- 
rently more general form Jl A° + V mB° + JnO° = 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 VI A 0 + Vm B° + VnG® = 0; and it is 
thereby very easily seen that the curve VIA’ + V m B u + V n ^ = 0 
can be represented by an equation VI' A' + Vm' B / + Vn'O ' = 0. 
But for v > 3, this is not so, and the curve VlA-\- &c. = 0 is only a 
particular form of the curve VIA’ + &c. = 0 ; 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 Jl A u + JnG’ = 0, and 
of the tetrazomal curve J A" + Jm B u + \/ nG° + JpD° = 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 Investigations; 
Part III., On the Theory of Foci ; and Part IV., On the Trizomal 
and Tetrazomal Curves, where the zomals are circles. There is, 
however, some necessary intermixture of the theories treated of, 
and the arrangement will appear more in detail from the headings 
of the several articles. The paragraphs are numbered continuously 
through the memoir. There are four Annexes, relating to questions 
which it seemed to me more convenient to treat of thus separately. 
