178 MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 



V X y will be the same at all the points of the curve ; wherefore 



y X 3/ = f ' X 3^' ; 

 and 



y=y' X- =asm0X-- 



Now, according to what was before shown, 



V = x/l+2<p(^), 



wherefore 



^="-^x\/Sif- (3.) 



By substituting this expression in the differential of the refraction, the problem will 

 be reduced to an integration. 



The equations that have been investigated are perfectly general, and will apply in 

 any constitution of the atmosphere that may be adopted. It has been thought better 

 to consider the manner in which the forces act, than to employ functions with pecu- 

 liar properties to express the molecular action. When the light in passing through 

 the atmosphere arrives at a surface of increased density, it receives an impulse which 

 may be considered as instantaneous ; and this impulse being distributed over the 

 breadth of a stratum of uniform density, ascertains the centripetal force tending to 

 the earth's centre, by the action of which the trajectory is described. 



3. It appears that Newton himself was the first to apply this new method to the 

 problem of the astronomical refractions. A table, which he had computed, and which 

 he gave to Dr. Halle y, is published in the Philosophical Transactions for 1721. 

 Nothing is said as to the manner in which the table was constructed : and it has 

 always been a curious and interesting question among astronomers, whether it is the 

 result of theory, or is deduced from observations alone, without the aid of theory. 

 Some original letters of Newton to Flamsteed, published in 1835 at the expense of 

 the Board of Admiralty, have put an end to all doubts on this point. These letters 

 prove that Newton studied profoundly the problem of the refractions ; that for several 

 months in succession he was occupied almost entirely in repeated attempts to over- 

 come the difficulties that occurred in this research ; during which time he calculated 

 several tables with great labour, namely, the one he gave to Halley, and another, or 

 rather three others, which have been found in his letters to Flamsteed lately printed. 



Admitting that the refractive power of the air is proportional to its density, which 

 Newton had established in his Optics on speculative principles, and which Hawksbee 

 afterwards was the first to demonstrate experimentally, the mathematical solution of 

 the problem requires a knowledge of the law according to which the densities vary 

 in the atmosphere. In his first attempt Newton assumes that the densities decrease 

 in ascending in the same proportion that the distances from the earth's centre in- 

 crease. Now a and r denoting the same things as before, put / for the total height 



