THE CYCLIDE. 



147 



since P and Q are any points on the conies, the whole system of circles will 

 form a cyclide. 



The circle corresponding to P a fixed point on the ellipse, is in a plane 

 which cuts that of xz along the line 



br 

 x = ~, 2/ = (9), 



and makes with it an angle 



The circle corresponding to Q a fixed point in the hyperbola, is in a 

 plane which cuts that of xy along the line 



* = f * = (10). 



and makes with it an angle 



Hence the planes of all the circles of either series pass through one of two 

 fixed lines, which are at right angles to each other, and at a minimum distance 



The line of intersection of the planes of the two circles through the point 

 R will therefore pass through both the fixed lines at points S and T, where 

 the co-ordinates of S are 



/*1* if ft" l* 



x = c r > y = ( c ) iana> z = (11), 



b o 



and those of T, 



x = , y = Q, z= sinB (12), 



c c 



and it is easily shewn that 



1 - T sec a 



o^t o ,, Q \ 



TR = r P (13) " 



1 cos B 

 c 



Hence, we deduce the following : 



192 



