546 Proceedings of the Royal Society 
the case of its summit being in oq on the hyperbola, when it will 
be of the form 
Y*o(p-<r 1 )-#( p -<r 1 )*=o, 
where 
8— - i 4- Sfoj -, + k — S*V, , 
0 1 c 
and p being now the vector of any point on the line <r 2 - oq or its 
prolongations. 
The last part of Minding’s propositions is proved by the com- 
parison of the values of the focal distances of the hyperbola and the 
ellipse, these being 
Jb 2 + c 2 = a, and Ja 2 -c 2 -b 
respectively, with the corresponding great axes, these being 
b and a . 
It follows also, from the expressions (44) of the vectors of the 
hyperbola and the ellipse, that the planes of these curves are at 
right angles, the line of intersection of the planes passing through 
the centre of these concentric curves. 
§ 9 . 
We will finally show that the variable vectors oq , cr 2 , are very ■ 
convenient auxiliaries for expressing the solutions of the equation 
of the fourth degree on which It depends, namely, of the equation 
(36) from which g has been eliminated by the help of (36 bis). 
We will not treat of that equation, but we consider the vector 
equation (3*7), from which it can be deduced as one of its scalar 
equations. 
Let p designate a point on the line a\, - cr l or its prolongations, 
so that we have 
p = oq + h't . 
We have then the two vector equations 
P ~Y 1 = V - 1 = I ix ( h ~ + i y( J l ~ 
+ k{z(h - M ^) + gab} 
