70 * 



The Kev. Charles Graves communicated the following ele- 

 mentary geometrical proof of Joachimsthal's theorem. 



Lemma 1. — If tangent planes be drawn at two points, P, P', 

 on a central surf ace of the second order ; and if perpendiculars be 

 let fall from, tlie points of contact on these tangent pilanes ; theper- 

 pendiadars will be proportioned 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 tico 

 tangent planes, and let the point S be taken on it so that the lines 

 PS, PS, 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 S has been taken in L, L', so that the angles 

 PSL, P'SL', are ecpjal, the point S will be that the sum of 

 whose distances from P and P' is a minimum. 



Again, the lines PS, PS, 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 parallel 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. 



