78 



ijfiilR)rhir>\^-TlS^mV^^ lo^ftti of the integralis that used 

 by Bradley. ■ •■"'•' >•'■->» JVivy*;. :>..» i.-.^ ^ 



Laplace uses the sahfie' liietbod of oblkfnTng the fluxional 

 equation as Simpson had done, and then proceeds to investi- 

 gate tho laws of reflection and refraction. He deriveb y an 

 analytical process . the conclusions, which Newton had de 

 (luced in the 14th section of the first book of the Principia. 

 Laplace next dtrrivGs his fundamental fluxional expression for 

 refraction which he shews may be integrated as far as 

 74° from the zenith, without a knowledge of the variation of 

 density in the atmosphere. 



In this paper the same fluxional expression, that Laplace 

 obtained, is deduced by a very short method, and by using 

 the common principle of the given ratio of the sines of in- 

 cidence arid refraction. Besides the simplicity of the inves- 

 tigation it has the advantage of avoiding hypothetic prin- 

 ciples lespecting the rays of light. 



The integration of the fluxional expression is also obtained 

 by a method that may be considered as entitled to notice. If 

 the surface of the earth were a plane, then whatever the law of 

 variation of the densities of the different strata of air parallel 

 thereto might be, the refractiou for any zenith distance 

 would be simply found from the knowledge of the refractive 

 force at the surface, by tbe constant ratio of the sines of in- 

 cidence and refraction. By the method given this part is 

 separated from the rest, and the effect of the spherical form 

 of the atmosphere is shewn. The formula for refraction 



