PROFESSOR CAYLEY ON POLYZOMAL CURVES. 51 



where k is the squared distance of the foci A, B, = 4aV suppose : whence putting 

 a?(\ — e 2 ) = ¥, the equation becomes 



A x + B x + 2\/A^B~ 1 - 46V = , 

 that is 



V\ + Vb; + 2bz= 0, 



which is the required new form. It is hardly necessary to remark that the 

 equation 2az + a/A + VE = 0, putting therein z = 1, and expressing A, B in 

 rectangular co-ordinates measured along the axes, is the ordinary focal equation 

 2a = a/(# — aef + f + </(as + aef + y\ 



117. I remark that the equation 2az + \ / A + \ / B = 0, gives rise to 

 4aV + A — B + 4#2-\/ A = 0, but here A — B = — laexz, so that the equation 

 contains z = 0, and omitting this it becomes (az — ex) + VA = 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. 



Tlieorem of the Variable Zomal as applied to a Conic — Art. Nos. 118 to 123. 



118. The equation 2kz + V^A + VB° = 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 Zkz + v^A^ — */W = 0, 2kz — Vh 5 + a/B° = 0, each ideally containing 

 (z = 0) the line infinity, and the order is thus reduced to be = 2. Each of the 

 circles A° = 0, B° = 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 + */A + VB = , 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 



2az + */(% — aezf + y 2 + V(x + acz) 2 + if = , 



the equation of the variable zomal or circle of double contact may be taken to be 



4:a 2 z 2 (x — aez) 2 + y 1 , (x + aezf + y 2 _ ^ 



-2 1-a 1+q 



where q is an arbitrary parameter ; writing for greater simplicity z — 1, and re- 

 ducing, the equation is 



(x — qae) 2 + y 2 = b 2 (l — (f) . 



120. If q ^ 1, then writing q = sin 6, we obtain the ellipse 



&_ . t. - 1 



a 2 ^ b 2 ~ ' 



