IV. GEOMETRICAL PROPOSITIONS APPLIED TO THE 

 WAVE THEORY OF LIGHT. 



[Transactions of thf Royal Irish Academy, VOL. xvn. Read June 24, 1833.] 



PART I. GEOMETRICAL PROPOSITIONS. 



1. THEOREM I. Conceive a curved surface B to be generated 

 from a given curved surface A in the following manner : having 

 assumed a fixed origin 0, 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 -R, so that OP and OR 

 may be reciprocally proportional to each 

 other, their rectangle being equal to a Fi g- 9 - 



constant quantity &% and let all the points R taken according 

 to this law generate the second surface B. Then the relation 

 beticeen these two surfaces, and between the points Q and R, will be 

 reciprocal ; that is to say, if a tangent plane be applied at the 

 point R 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 rect- 

 angle under them being also equal 7r. 



2. To prove this theorem, take a point q, in the tangent plane 

 of the surface A, and near the point of contact Q (Fig. 9). 

 Through q let several other planes be drawn touching the sur- 

 face A in points Q', Q", Q'", &c., and draw the perpendiculars 

 OP'Rf, OP"R", OP"'R"\ fec., according to the same law as 

 OPR. The points R, Rf, R", Rf", &c., will thus be upon the 

 second surface B, and they will moreover be all in the same 



