; Geometrical Propositions applied to the WVave Theory of Light, 
. By James M‘Cuxxacu, F.T.C.D. 
¢ : * 
Read June 24, 1833. 
Part I.—GeromerricaL Propositions. 
1. Turorem I. Conceive a curved surface B to be generated from a given curved sur- 
~~ face Jin the following manner : having assumed a fixed origin O, apply a tangent plane 
at any point Q of the given surface, and perpendicular to this plane draw a right line 
OPR cutting the plane in P, and terminated in RF, so that OP and OR may be re- 
ciprocally proportional to’each other, their rectangle being equal tp a constant quan- 
tity k°, and let all the points & taken according to this law generate the second sur- 
face B. Then the relation between these two surfaces, and between the points Q and 
_ R, will be reciprocral ; that is to say, if,a tangent plane be applied at the point F of 
the second surface, a perpendicular ON to this plane will pass through the point Q of 
the first surface, and ON and OQ will be reciprocally proportional to each other, the 
rectangle under them being also equal to k*. 
2. To prove this theorem, take a point g, in the tangent plane of the surface 4, 
_ and near the point of contact Q. (Hig. 2.) Throughg let several other planes be 
drawn touching the surface A in points Q’, Q", Q’, &c.. and draw the perpendi- 
 culars OP'R’, OP"R", OP" R”, &e. according to the same law asOPR. The 
points R, R’, R", RK”, &c., will thus be upon the second surface B, and they will 
moreover be all in the same plane ; for from any one of them F' let Fn be drawn 
perpendicular to the right line Og and meeting Og in 7; then on account of the si- 
ilar right-angled triangles OP'q and On’, the rectangle  Oq will be equal to the 
rectangle 2’ OP’, or to the constant quantity k*, so that the point x, or the foot 
_ of the perpendicular let fall upon Og, will be the same for all the points R,R'.R’,R”, 
_ &c., and consequently all these points will lie in a plane cutting the right line Ogn 
perpendicularly in , so as to make the rectangle n Oq equal to k*. Now while the 
point Q remains fixed, let the point g approach to it without limit in the tangent 
