of Edinburgh, Session 1881-82. 
547 
Taking the scalars S . i , S . j , S . k , we get six equations, which 
form only four distinct ones, because they satisfy a priori to 
h' 2 = - 1 , and to 1ST 2 = - (c r 2 - oq) 2 . From these equations we 
easily deduce 
68 ><*(£) 
and also 
FT 
= S 
p-<r 1 
°2 “ °"] 
where we employ the sign S (placed before the quotient of two 
parallel vectors) simply for preserving the algebraical sign of the 
quotient. 
If we assume oq and <r 2 to be given, then the expression 
(46) h = P + c 2 S£^l. 
°*2 _ °T 
will represent the values of h for any point p situated on the line 
<r 2 - oq , and the coordinates x, y , 0 of that point, together with 
the resulting value of h, will satisfy the equation of the fourth 
degree spoken of. 
If we assume p to be given, namely, x , y , z to be given, then 
the question would be to determine first oq and o* 2 independently 
of h, and this would again lead to an equation of the fourth degree 
in one of the coordinates x x , z 1 , or y 2t z 2 , and the question as to 
the reality of the roots would still arise in all its complication. 
Instead of following that course, we may by geometrical con- 
siderations, and with the help of (46), assure ourselves that for 
every point of space there exist four corresponding lines <r 2 - oq , 
and consequently four values of h which satisfy to the equation in 
Jc , and consequently to the scalar equation deduced from it. 
Let us suppose that the two Minding’s curves be constructed in 
their relative position in space, and let them consist of a rigid 
material. Then drawing a straight line through the extremity M 
of p we may turn that line until it strikes, say, one branch of the 
hyperbola, then we move the line on, without detaching it from 
the hyperbola, until it strikes a point E of the ellipse, the point of 
the hyperbola being H. Then we have a first position of the line 
<r 2 - oq , and a corresponding value of h and of k!. 
VOL. XL 3 Y 
