235 
By ascribing to n the values 1, 2, 3, &c., we shall obtain 
a series of ,elli]3ses of which the areas ai'e in geometrical 
progression, each bearing to the next the ratio m}. The 
general equation will also include the primitive if to n the 
value 0 be ascribed ; the sum of all these areas (including 
• • • • TTC^ • • 
the primitive) will be if we suppose the operation to be 
repeated an infinite number of times. The semi axis-major 
of any ellipse will be 
v/ 
^ 1 - 2m^m-'^ + - (1 
s + 
,-1 
1 
r— 1 
> 
g 
1 
semi axis-minor 
/t i O O,. 
■j' 
(1 + 
The inclination (0) of the major axis to the axis of x is given 
by the equation 
_ ^ml 
taii20 = 2 ^ 
This equation may also be written in the form 
^ tan2ri) 
tan20 = ^ 2 » o ; 
1 + m sec20 
9 denoting the inclination of the primitive axis to the axis 
of X. 
If we suppose n to become infinite, .we shall have 
utimately 0 = 0, the minor axis will have the value 0, and 
the major axis will assume the ambiguous form on exami- 
2c 
nation this will be found to have the value y. To find this 
limit geometrically, draw tangents to the circle at the 
points 2 / = 0, x = c and y = 0x- -c and produce the primi- 
tive axis both ways, then the portion intercepted by these 
tangents will be the limit of the ellipse. 
We may give an extension of meaning to (2) by making 
