to the Wave Theory of Light. 31 



apsidal surfaces, from the manner in which they are conceived to 

 be generated. 



Let G be a given surface, and a fixed origin or pole. If 

 a plane passing through cut the surface G, the curve of inter- 

 section will in general have several apsides A, A.', A." ' , &c., where 

 the lines OA, OA, OA", &c., are perpendicular to the curve. 

 Through the point conceive a right line perpendicular to the 

 plane of the curve, and on this perpendicular take from the 

 distances Oa, Oa', Oa", &o., respectively equal to the apsidal dis- 

 tances OA, OA', OA", &c. Imagine a similar construction to be 

 made in every possible position of the intersecting plane passing 

 through 0, and the points a, a', a", &o., will describe the dif- 

 ferent sheets of an apsidal surface. 



The apsidal surface has a centre at the point 0, because the 

 lengths Oa, Oa', Oa", &o., may be measured on the perpendicular 

 at either side of the intersecting plane. 



Referring* to the demonstration of Theorem III., it will be 

 seen to depend only on the supposition that the point Q is an 

 apsis of the section made by the plane QOq ; or, which is the 

 same thing, that OQ is a position wherein the radius vector from 



to the curve of section is a maximum or a minimum. Hence 

 we have the following general theorem : 



24. PROP. YI. THEOREM. If tangent planes be applied at 

 corresponding points A, a, on the surface G and the apsidal sur- 

 face which it generates, these tangent planes will be perpendi- 

 cular to each other and to the plane of the points 0, A, a. 



This is equivalent to saying that perpendiculars from on 

 the tangent planes are equal to each other, and lie in the plane 

 of the lines OA, Oa. 



25. If Q and R be reciprocal points on two reciprocal surfaces, 

 of which Q is the fixed origin or pole, the tangent plane at Q 



Professor Lloyd. See Transactions of the Royal Irish Academy, Vol. xvii., part i., 

 pp. 132, 145 ; and the present Paper, Art. 55-58. 



The indeterminate cases of circular section at least the case of the nodes had 

 occurred to me long ago ; but having neglected to examine the matter attentively, 



1 did not perceive the properties involved in it (13). 



* Transactions of the Royal Irish Academy, Vol. xvi., part ii., p. 68. 



