224 
Proceedings of Royal Society of Edinburgh. [sess. 
provided the axis of p be moved upwards parallel to itself through 
-a (^stance = v. 
The problem may now be restated thus : it is required to find 
that point in the p - v plane whose ordinate to the most distant of 
the three surfaces is a minimum. 
This, a problem in three dimensions, may be reduced to a 
problem in two only, by expressing it in the following form, which, 
Fig. 10 . — In drawing the above figure, 1 teas taken — 2 and r = l , for 
the sake of clearness. Such a combination of values would never 
of course represent an actual case. 
from the foregoing discussion of fig. 10, can easily be seen to be 
its equivalent : find that position for the p axis of fig. 10, supposed 
moved parallel to itself, and that value of p , such that the ordinate 
•of the latter to the most distant of the three curves (the hyperbola 
and the two straight lines) is a minimum. 
A little consideration will show that the solution is as follows. 
Move up the p axis through such a distance that it bisects the 
shortest vertical line that can be drawn, terminated at one end by 
