310 TRANSACTIONS OF SECTION A. 
infinity. the other three lie on the straight line x = 3/8, which may be regarded as a 
circle of infinite radius ; the w- axis may be regarded as two coincident asymptotes. 
(4) When 1 >e>4, there are two asymptotes making equal angles with the x- axis. 
As eé increases from 4 to 1 the angle between the asymptotes increases from 0° to 180°, 
their intersection, V, moves from (—oo, 0) to the origin; the cusp C’ precedes V 
from (—co, 0) to (0, 0); the distance D D’ increases to a maximum 44/3/9 when 
e = /(2/3), and then dwindles to 0. 
(5) When e =1, the Limacon is a cardioid, the evolute is also a cardioid ; it passes 
through the origin, its cusp C is at (2/3, 0); the three cusps C’, D, D’, coincident at 
the origin, become a simple point on the cardioid. 
(6) When e>1, the cusps D, D’ have disappeared, the cusps C, C’ point inwards, 
the curve is closed, the shape of the curve does not change much as e increases, the 
distance C C’ diminishes from 2/3 to 4, the total height (parallel to the y- axis) increases 
from 4/3/2 to 1, the breadth, now less than the height, diminishes from 3/4 to 1/2. 
(7) When c=, the evolute is altogether at infinity; the difference c—c’ is 
found to be finite and equal to 4, likewise the height, 1, the breadth, 1/./2; the position 
of the breadth, 3/./2, above and below C C’ enable an illustrative figure to be drawn. 
It is symmetrical about its central ordinate. 
Diagrams are given to illustrate these cases, except (1), which needs no diagram. 
(2) e=°3, the star elongated. 
(3) ¢ = 4, with three cusps in a straight line. 
(4) e=43, with the four cusps at the corners of a square. 
(5) e=1, where three cusps are hidden in one point. 
(6) e=2, the evolute of the Trisectrix. 
(7) ¢ = 9, with double symmetry. 
é€ = co 
The equation of the evolute is 
i { P+ (e—e) (a—c’) \ "4.276%? (y2+a®—eax)?=0. 
5. On the Algebraic Theory of Modular Systems (or Modules of 
Polynomials). By F. 8. Macauuay, M.A. 
The theory of modules of polynomials is still only in its infancy, although 
it may claim an age of 185 years, its origin dating from the ‘Théorie 
Générale des Equations Algébriques,’ par M. Bézout, de l’Académie Royale des 
