548 Proceedings of the Royal Society 
But we remark that if the points are following each other in the 
order say 
M, H, E, 
there will exist another position of the line in which it strikes the 
ellipse first, and further on the hyperbola, so that we have a second 
line cr 9 — o" 1 , corresponding to the disposition of either 
M, E', H\ 
or, may he, if the first had been 
H, M, E, 
then there would be a second disposition 
M, H', E'. 
But we may repeat this reasoning in taking into account the 
other branch of the hyperbola, and so we get four systems of posi- 
tions of the line <r 2 - oq and four corresponding values of h , these 
being real of necessity, because their expression is dependent on the 
real lines p -.oq , and <r 2 - oq in 
7» = &2 + c 2 S-^j-. 
<r 2 oq 
We see also by this expression that the points corresponding to 
the value h = a 2 can be placed only on one of the cylinders (40), 
because, apart from the value p = oq , which places p on the inter- 
section of both the cylinders, we have also the points p for which 
we have generally 
S^-l, 7i = Z> 2 + c 2 = a 2 
<Tl 2 °”l 
As p - oq and o- 2 - oq are to have the same direction, the condition 
can only be satisfied (excepting always p = oq) when oq becomes 
infinite along the hyperbola. In this case p — <r 1 takes the direction 
parallel to the asymptotes, and therefore parallel to the generating 
line of one of the cylinders (40). It follows that, for h — a 2 , the 
only surface represented by the primitive equation (4) are the 
cylinders (40). 
