236 
n negative. The equation may then he written in the form 
by giving to n the values 1, 2, 3, &c., a series of ellipses 
will be obtained of which the areas are still in geometric 
progression, but now the series is divergent, each area 
having to the following the ratio m~^ . The circle which, 
when n was positive, was regarded as the primitive curve, 
may now be regarded as the ultimate product of a series of 
operations which may be carried back as far as we please. 
The semi-axes of the '^th ellipse will be 
J2,lc 
— ± (1 - 
Also the inclination d of the minor axis to the axis of x will 
be given by the formula 
tan2(^m^'' 
taii20 = 
+ sec2(/) 
(j) denoting the inclination of the primitive axis to the axis 
of X. If n be made infinite 0 = 0, the minor axis becomes 
I c and the major axis infinite, the points of contact with 
the lines x-c and x= -c are situated at an infinite distance. 
As previously stated, 
we have some choice 
of method in piling one 
on another, the elemen- 
tary parallelograms ob- 
tained by the compo- 
sition of projections. A 
more general expression 
for the equation of the 
generated curve may be 
obtained as follows : 
Let I and m be the 
cosines of the inclination of the primitive AB axis to 
the axis of x and y, draw any arbitrary curve HK ter- 
