720 PHILOSOPHICAL TRANSACTIONS. [aNNO 1790. 



that all the parts of light are equally acted on in their passage by bodies ; and 

 deduce several of the most important propositions which occur, without men- 

 tioning the demonstrations. 



Def. 1. If a ray passes within a certain distance of any body, it is bent in- 

 wards ; this we shall call inflection. 2. If it passes at a still greater distance it is 

 turned away ; this may be termed deflection. 3. The angle of inflection is that 

 which the inflected ray makes with the line drawn parallel to the edge of the 

 inflecting body, and the angle of incidence is that made by the ray before 

 inflection, at the point where it meets the parallel. And so of the angle of 

 deflection. 



Prop. 1. The force by which bodies inflect and deflect the rays, acts in lines 

 perpendicular to their surfaces. 



frnp. 2. The sines of inflection and deflection are each of them to the sine 

 of incidence in a given ratio ; and what this ratio is we shall afterwards show. 



Prop. 3. The bending force is to the propelling force of light, as the sine of 

 the difference between the angles of inflection, or deflection, and incidence, to 

 the cosine of the angle of inflection, or deflection. 



Prop. 4. The rays of light may be made to revolve round a centre in a spinal 

 orbit. 



Prop. 5. If the inflecting surface be of considerable extent, and a plane, then 

 the curve described may be found by help of the 41 Prop. Book 1, Principia ; 

 provided only, the proportion of the force to the distance be given. Thus, 1. 

 When the bending force is inversely as the distance, the curves to be squared 



are, a conic hyperbola, and a logarithmic, t/" = -. The trajectory, therefore. 



It 



cannot be found in finite terms ; its equation is //^ / — = ,r ; and tlie sub-tan- 

 gent is to the sub-normal as 1 to / -. 2. When the bending force is in- 

 versely as the square of the distance, the curves to be squared are a cubic hyper- 



1 • r - 



bola, 7/ = — , and a conchoid, y"^ =i — ^ — ; therefore the equation to the trajec- 

 tory (« — x) if =. xx'' ; which belongs to a cycloid, the radius of whose gene- 

 rating circle is a. In general, if the force be inversely as the mth power of 

 the distance, the equation of the trajectory will be (a'"~' — x'"~') ij" = .z'"'~'.f'; 

 which agrees also with the first case, where m being = 1 o""', may be esteemed 

 the hyp. log. of a. If the force be inversely as the cube of the distance, the 

 curve is a circular arch, and that of deflection is a conic hyperbola. (Principia, 

 lib. 1, prop. 8). If the inflecting body be a globe or cylinder, and the force 

 be inversely as the square of the distance from the surface, then by prop. T\, 

 book 1, Prin(i[)ia, the attraction to the centre is inversely as the square of the 



