ON POLYZOMAL CURVES. 
519 
414] 
where k is the squared distance of the foci A, B, = 4a 2 e 2 suppose: whence putting 
a 2 (l — e 2 )=b 2 , the equation becomes 
% + 23 x + 2 VCi; - 46V = 0, 
that is 
+ V^ + 26* = 0, 
which is the required new form. It is hardly necessary to remark that the equation 
2az + V2i + V23 = 0, putting therein z—1, and expressing 21, 23 in rectangular coordinates 
measured along the axes, is the ordinary focal equation 2a = f(x — ae) 2 + y 2 + f(x + ae) 2 + y 2 . 
117. I remark that the equation 2a2 + V2l+V23 =0 gives rise to 4a 2 z 2 + 21 — 23 +4a^V21=0, 
but here 21 — 23 = — 4aexz, so that the equation contains z = 0, and omitting this it 
becomes (az — ex) + V21 = 0, a bizomal form, being a curve of the order = 2, as it should 
be; this is in fact the ordinary equation in regard to a focus and its directrix. 
Article Nos. 118 to 123. Theorem of the Variable Zomal as applied to a Conic. 
118. The equation 2kz + V2l° + V23 0 = 0 is in like manner that of a conic; in 
fact, this would be a curve of the order -- 4, but there are as before the two branches 
2kz + V2l° — V23° = 0, 2kz — V21° + V23° = 0, each ideally containing (2 = 0) the line infinity, 
and the order is thus reduced to be = 2. Each of the circles 21° = 0, 23° = 0 is a 
circle having double contact with the conic (this of course implies that the centre of 
the circle is on an axis of the conic). We may if we please start from the form 
2kz + V21 + V23 = 0, and then by means of the theorem of the variable zomal introduce 
into the equation one, two, or three such circles. 
119. It is in this point of view that I will consider the question, viz., adapting 
the formula to the case of the ellipse, and starting from the form 
2 az + V(ic — aez) 2 + y 2 + f(x + aez) 2 + y 2 = 0, 
the equation of the variable zomal or circle of double contact may be taken to be 
4aV (x — aez) 2 + y 2 (x + aez) 2 + y 2 _ 
-2 + 1 -q + 1+2 “ ’ 
where q is an arbitrary parameter; writing for greater simplicity z — 1, and reducing, 
the equation is 
(x — qae) 2 + y 2 = b 2 (1 — q 2 ). 
120. If 2<1, then writing <7= sin#, we obtain the ellipse 
x 2 y 2 
a? b 2 
1, 
as the envelope of the variable circle 
(x — ae sin 6) 2 +y 2 =b 2 cos 2 6,