234 
the same paper it was pointed out that the projectrix might 
itself in turn he regarded as a primitive, and that by a 
repetition of the operation it would yield a secondary pro- 
jectrix, and so the process might be carried on indefinitely. 
As in the paper referred to, let the primitive axis make with 
the axes of x and y angles of which the direction cosines are 
I and m ; to simplify the equations, suppose that the primi- 
tive axis passes through the origin so that the equation is 
m 
As a particular and simple example, let the primitive curve 
be the circle 
X=^ + Y2 = c^ (1) 
Proceeding in the same manner as formerly described, the 
elementary parallelograms obtained will be piled one on 
another, so that their centres lie on the primitive axis. 
X and Y being the co-ordinates of a point on the primi- 
tive, and X and y the co-ordinates of the corresponding 
point on the projectrix, we have the relation 
y ~ m(/X + mY) 
Substituting in ( ) we obtain the equation 
{y - mlxf = m\c^ - 
representing an ellipse. If this curve be regarded as a 
primitive we shall obtain from it by a repetition of the 
operation 
(y - mlx{l + = m8(c^ - x‘^)y 
and if the operation be repeated n times the nth. projectrix 
will be 
[ y - mlx{ 1 + 4 - + etc ^ - x'^) 
Since m is generally less than unity, the co-efficient of the 
term mix will be a converging series of which the sum is 
^ ■ substituting in the equation and remembering that 
1 - 
= l we get the equation 
( 2 ) 
