70 
The Rey. Charles Graves communicated the following ele- 
mentary geometrical proof of Joachimsthal’s theorem. 
Lemma 1.—/Jf tangent planes be drawn at two points, P, P’, 
on a central surface of the second order ; and if perpendiculars be 
let full from the points of contact on these tangent planes ; the per- 
pendiculars will be proportional to the perpendiculars let fall from 
the centre of the surface upon the tangent planes. 
This is evident in the case of the sphere; and the theorem 
may be extended to the other surfaces by a simple deformation. 
Or it may be proved analytically in the simplest way, by means 
of the ordinary equation of the tangent plane. 
Lemma 2.—Let LL’ be the line of intersection of the two 
tangent planes, and let the point S be taken on it so that the lines 
PS, P'S, make equal angles with the line LL’; then the lines 
PS, P'S, will be reciprocally proportional to the perpendiculars 
let fall from the centre upon the tangent planes at P and P’. 
For the lines PS, P’S, are evidently proportional to the 
perpendiculars let fall from P, P’, upon the tangent planes ; 
and these, by the preceding Lemma, are proportional to the 
perpendiculars let fall from the centre upon the tangent planes 
at P’ and P. 
If the point 5S has been taken in L, Li, so that the angies 
PSL, P’SL, are equal, the point S will be that the sum of 
whose distances from P and P’ is a minimum. 
Again, the lines PS, P'S, being tangents, are proportional 
to the parallel semi-diameters of the surface. We may, there- 
fore, state the result at which we have now arrived in the fol- 
lowing proposition. 
If two points on a central surface be connected by a shortest 
line passing over the line of intersection of the two planes which 
touch the surface at those two points; the semi-diameters of the sur- 
face paralles to the two straight portions of the shortest line will 
be reciprocally proportional to the perpendiculars let fall from 
the centre upon the tangent planes in which those portions are 
respectively contained. 
