the vectorial equations of confocal orthogonal bicircular quartics, each of the form

lp + mp' + np′′ = 0;

with similar equations of the same curves, connecting r1, r,, r, and

Hence the theorems are proved (Salmon, Higher Plane Curves, Chap. VI.) that given four concyclic foci of a bicircular quartic, two such quartics can be described through any point, and these cut each other at right angles."

Also, if any cylinder, whose cross section is a bicircular quartic, be electrified, the electrification at any point will be inversely proportional to the square root of the product of the distances from the four real foci.

All these theorems are obtained immediately by the inversion of the system of confocal Cartesians of § 2, three foci of the inverse system of curves, the inverse points of the foci of the Cartesians lying on a circle passing through the centre of inversion, the fourth focus being at the centre of inversion; and then the vectorial equations are obtained by elementary geometry.

10. When two foci B, C of the three foci A, B, C of a system of confocal Cartesians coincide, then k=1, and we obtain the systems of confocal limaçons,

where is the distance of a point P from the double focus O, r' from the single focus A, and a is the distance between the foci. Denoting the angle POA by 0, then

If the foci B, C, after coincidence, now move at right angles to the original line of foci, so that ABC is an isosceles triangle, we obtain the system of curves derived from the integral

n

the elliptic functions of being now to the complementary modulus k', and by the alternate elimination of and n we obtain the vectorial equations of the orthogonal quartic curves, having foci at A, B, C.

11. Inverting the previous system of curves with respect to any point on the line through A perpendicular to BC, we obtain the system derived from the integral

having four foci A, B, C, D forming a kite-shaped figure; and

then

and by the alternate elimination of έ and n, we obtain the vectorial equations of the orthogonal quartic curves, having foci at A, B, C, D.

When ABCD is a rhombus, the system of curves is as in § 5, given by

x+iy=cn (§ + in).

Generally if we invert the system of curves of § 10 with respect to any point whatever, we obtain the second class of bicircular quartics, in which the four real foci do not lie on a circle, but are so related that

AB: BD :: AC: CD.

12. In order to solve the hydro-dynamical problems of determining the current and velocity functions of liquid motion due to the motion of cylinders whose cross-sections are curves § constant, or n=constant; taking for example the confocal Cartesians defined by

x + iy = sn2 (§+in) ;

and therefore is the current function for the motion of the liquid between the cylinder, na and n=6, when the cylinder na is moving parallel to the axis of a with velocity U and the cylinder = 8 is fixed.

If denote the velocity function of this motion, being the conjugate function to ↓,

In a similar manner the current and velocity functions due to any motion of translation or rotation of any of the cylinders, = constant, or constant, may be written down.

=

(2) On a Mathematical Law of Interest in Political Economy. By Dr AKIN-KAROLY.