3. To determine the equation to a non-singular cubic of the third order.

By interchanging the variables p, q, r, with P, Q, R in the resulting equation of §2, we may obtain the required equation to the evolute of the cubic

aa +30 + c +2day=0.

The principle of the process is best seen in Spherics, where both the primitive and reciprocal curves are finite.

4. To investigate the Boothian equation to the evolute of a non-singular cubic of the third class.

The equations to the curve and a point of contact, and the condition of perpendicularity of the normal, are

do df

$ (f, 9) = 0, (§ − ƒ) aƒ + (n − 9 )

do

=0, fe+gn=0.

dg

Their eliminant determines the evolute.

A non-singular cubic of the third class is thus denoted:

af3 + bg3 +c+6dfg = 0,

The equation to its evolute is

4d3 (§* — n2)3 — 3cd§n (§3 — n2) (b§ + an) + ac2n3 − bc3ç3

× √ {d* (§3 − n3)2 + 2cd§n – c§n (an+b§)}.

--

This may be otherwise expressed:

(a§3 + bn3 + c + 6d§n) {b2c3ç3 – 2 (abc + 16d3) §3n3 + a3cn®}

= (§2 + n2)2 (8d3 + abc) {− 6d§3n2+ an3 (n3 — 2§3) + b§3 (§3 — 2n3)}.

=

Cheltenham, June 1, 1870.

(To be continued).

VOL. XI.

G

NOTE ON A RELATION BETWEEN TWO CIRCLES.

By Professor CAYLEY.

CONSIDER any two circles O, Q; and let AC, BD, A'D', B'C' (fig. 12) be the common tangents touching the circles in the points A, A', B, B', C, C', D, D': the locus of a point P such that the pairs of tangents from it to the two circles respectively form a harmonic pencil, is a conic through the 8 points A, A, B, B', C, C', D, D'; but this conic may break up into two lines, viz., if (as in the figure) the points A, B', D', D are in a line, then the points C, C', A', B, will be in a symmetrically situated line, and the conic breaks up into this pair of lines, meeting, suppose in K. The condition for this, if a, a' are the distances of the centres from a fixed point in the line of centres, and if the radii are c, c' is readily found to be

Suppose in general, that (given any two conics) the point P' is the intersection of the polars of P in regard to the two given conics respectively; then if P describes a line, the locus of P' is a conic passing through the three conjugate points of the given conics; if, however, the line which is the locus of P pass through one of the conjugate points, then the conic the locus of P breaks up into a pair of lines, one of them a fixed line through the other two conjugate points, the other of them a line through the first mentioned conjugate point. That is, if the locus of P be a line through a conjugate point, the locus of P' is a line through the same conjugate point; but in every other case the locus of P' is

a conic.

Reverting to the figure of the two circles, in order that it may be possible that the two lines AD and BC may be loci of points P, P', related as above, it is necessary that K shall be a conjugate point of the two circles; that is, if the two circles intersect in points A, A' lying symmetrically in the radical axis, which meets, suppose, the line of centres in M, then it is necessary that K shall be one of the anti-points of A, A'; or what is the same thing, the

=

distance KM must be into MA or MA'; this condition, if as above (a-a')2 = 2 (c2+c"), implies c2=c", and we have then (aa)=4c", that is, the circles must be equal, and the distance of the centres must be twice the radius, or what is the same thing, the circles must be equal circles touching each other; when this is so, the two lines AD, BC being then lines at right angles to each other intersecting in the point of contact, have, in fact, the above mentioned relation. And it thus appears that given two circles, the necessary and sufficient conditions for the coexistence of the properties mentioned in the theorem are that they shall be equal circles touching each other.

ON THE PORISM OF THE IN-AND-CIRCUMSCRIBED POLYGON, AND THE (2, 2) CORRESPONDENCE OF POINTS ON A CONIC.

By Prof. CAYLEY.

THE present paper includes, as will at once be seen, much that is perfectly well known; but the separate theories required, it seemed to me, to be put together; and there are, particularly as regards the unsymmetrical case afterwards referred to, some results which I believe to be new.

The porism of the in-and-circumscribed polygon has its foundation in the theory of the symmetrical (2, 2) correspondence of points on a conic; viz. a (2, 2) correspondence is such that to any given position of either point there correspond two positions of the other point; and in a symmetrical (2, 2) correspondence either point indifferently may be considered as the first point and the other of them will then be the second point of the correspondence. Or, what is the same thing, if x, y are the parameters which serve to determine the two points, then x, y are connected by an equation of the form (*) (x, 1)2 (y, 1)*=0, which is symmetrical in regard to the two parameters (x, y). In the case of such symmetrical relation it is easy to show that the line joining the two points envelopes a conic. For the relation may be expressed in the form (*X1, x + y, xy)* = 0;

we may imagine the coordinates (P, Q, R) fixed in such manner that for the point (x) on the first conic we have P: Q: R=1:x:x2, and for the point (y), P:Q:R=1:y:y2; the equation of the line joining the two points is, then,

that is

P, Q, R = 0;

1, x, x2

1, y, y2

PxyQ(x + y) + R = 0,

or representing this by

we have

P§ + Qn+ R¢=0,

nxy:-x-y: 1; and consequently (, n, 5) are connected by a quadric equation; that is, the envelope

is a conic.

The relation (Xx, 1)2 (y, 1)2=0, whether symmetrical or not, leads as will be presently shown to a differential equation of the form

where X, Y are quartic functions of x, y respectively; viz., these are unlike or like functions of the two variables according as the integral equation is not or is symmetrical in regard to the two variables. In the former case, however, the functions X, Y are so related to each other, that the two can be by a linear transformation converted into like functions of the variables: for instance, if y be changed into ay, +b+cy, +d, then the constants may be determined in suchwise that Y is the same function of y,, that X is of x; the original integral equation being hereby converted into a symmetrical equation (Xx, 1) (y,, 1)2=0 between x and y1, so that in one point of view the unsymmetrical case is not really more general than the symmetrical one. It is to be added that the integral equation contains really one more constant than the differential equation (this is most readily seen in the symmetrical case, the differential equation depends only on the ratio of five constants a, b, c, d, e, whereas the integral equation depends on the ratio of six constants) so that the integral equation is really the complete integral of the differential equation.

Attending now to the symmetrical case; if A and B are corresponding points, then the corresponding points of B are 4 and a new point C; those of Care B and a new point

D, and so on; so that the points form a series A, B, C, D... ; and the porismatic property is that, if for a given position of A this series closes at a certain term, for instance, if D=A; then it will always thus close, whatever be the position of A. And this follows at once from the consideration of the dx

=

dy

differential equation √(X) = √(Y); viz., as this is at once integrable per se in the form

this equation must be a transformation of the original equation (*Xx, 1)* (y, 1)2=0, and equally with it represent the relation betwen the parameters x, y of the two points A, B; the constant of integration & is of course completely determined in terms of the coefficients of the last mentioned equation, assumed to be given.

Hence forming the equations for the correspondences, B, C; C, D... and assuming that the series closes F, A; we have

where, however, the II (a) of the last equation must be regarded as differing from that of the first equation by a period, say 2, of the integral; hence adding, we have

which gives between the constants of the integral equation (*Xx, 1)2 (y, 1)2 = 0, a relation which must be satisfied when the series, closes at the nth term (viz. when the term after this coincides with the first term); and this relation is independent of x, that is, of the position of the point A.

The analysis in regard to the differential equation is as follows:

Consider the equation

U= y2 (ax2 +2bx + c)

+ 2y (a'x2 + 26'x + c')

+

(a"x2 + 26"x + c′′) = 0,

say

we have

U= (P, Q, RXy, 1)2 = (L, M, NXx, 1)2=0,

dU = 0 = (Py + Q) dy + (Lx + M) dx.