applied to the Wave Theory of Light. 24.9 
The apsidal surface has a centre at the point O, because the lengths Qa, 
Oa’, Oa’ ,&c. 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 Q Oq; or, 
which is the same thing, that OQ is a position wherein the radius rector from O to the 
curve of section is a maximum or a minimum. Hence we have the following general 
theorem :— 
24. Prop. VI. Tueorem. If tangent planes be applied at corresponding points | 
A, a,on the surface G and the apsidal surface which it generates ; these tangent 
planes will be perpendicular to each other and to the plane of the points O, A, a. 
This is equivalent to saying that perpendiculars from Oon 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 O is the 
fixed origin or pole, the tangent plane at Q will be (1) perpendicular to OF and to 
the plane QOR. Leta plane also perpendicular to the plane Q OF pass through OQ, 
cutting the surface to which the point Q belongs in a certain curve, and the tangent 
plane at Q in a tangent to this curve. The tangent is evidently perpendicular to Og, 
and therefore the point Q is an apsis of the curve. 
In like manner, the point £ is an apsis of the section made in the other surface by 
a plane passing through OF and perpendicular to the plane QOR. 
26. From these observations, and from Prop. VI., it appears that if the points 
Q, R, in the figure of Theorem IV., be reciprocal points on any two reciprocal sur- 
faces, and if the same construction be supposed to remain, the points 7’ and M will 
be points on the apsidal surfaces generated by these reciprocal surfaces, and the tangent 
planes at Z’and MM will be perpendicular to the lines OM and OT respectively. Also 
the rectangles LO Z' and MOS will be equal to &°. Hence we have another general 
theorem :—- 
Prop. VII: Turorem. The apsidal surfaces generated by two reciprocal surfaces 
_ are themselves reciprocal. 
27. A very simple example of apsidal surfaces, with nodes and circles of contact, 
may be had by supposing the generatrix G@ to be a sphere, and the pole O to be within 
the sphere, between the surface and the centre C. 
It is evident that the apsidal surface in this case will be one of revolution round the 
right line O Cas an axis. Therefore taking for the plane of the figure (Fg. 6.) a plane 
passing through OC'and cutting the sphere in a great circle of which the radius is C'S, 
let a plane at right angles to the figure revolve about O, cutting the circle C'S in the 
points 4, 4’. The section of the sphere made by the ~ailiving plane will have only 
* Transactions of the Royal Irish Academy, Vol. XVI. Part II. p. 68. 
