188 ON THE MECHANICAL DESCRIPTION OF A NODAL BICIRCULAR QUARTIC. [446
through a fixed point of the plane II. And secondly, if a fixed circle C\ in the plane Ili
always touches a fixed line L in the plane II, this is equivalent to the condition that
a fixed point in the plane n x is always situate in a fixed line L 1 in the plane n x .
The different forms of condition therefore are :
(a) A fixed circle C x in the plane always touches a fixed circle C in the
plane II (where, as above, either circle indifferently may be reduced to a point).
(/3) A fixed line in the plane Eh always passes through a fixed point G in
the plane II.
(7) A fixed point C x in the plane Eh is always situate in a fixed line L of the
plane II.
Hence, if the motion of the plane Hi satisfy any two such conditions (of the
same form or of different forms, viz., the conditions may be each a, or they may be
a and /3, &c.), then the motion of the plane n x will depend on a single variable
parameter, and the question arises as to the locus described by a given point, or
enveloped by a given line, of the plane n; and again of the locus traced out, or
enveloped, on the moving plane n x by a given point of the plane n. The case con
sidered in the present paper is of course a particular case of the two conditions being
each of them of the form a.
It may be remarked, that if the two conditions be each of them /3, then there
will be in the plane Hi a fixed point C x which describes a circle; and similarly, if
the two conditions be each of them 7, then there will be in the plane II x a fixed
point G x which describes a circle ( x ); that is, the combination ¡3/3 is a particular case
of a/3, and the combination 77 a particular case of ary.
1 The theorem is, that if an isosceles triangle, on the base A A' and -with angle =2w at the vertex C,
slide between two lines OA, OA' inclined to each other at an angle w, in such manner that G is the centre
of the circle circumscribed about OAA', then the locus of C is a circle having 0 for its centre.
