MR. IVORY ON THE THEORY OP ASTRONOMICAL REFRACTIONS. 17J> 



of the atmosphere ; then <p (g) the refractive power of the air at the distance r from 

 the centre of the earth, will, according to this hypothesis, be expressed by the formula 



9 (f ) = ? (f') X 2 . 



If this value be substituted in the formula (1.), which is a deduction from the sixth 

 proposition of the first book of the Principia, the result will be 



In this expression we have 



and as 2 9 (^), or the increment of the square of the velocity of the light is very mi- 

 nute, amounting to less than '0006 in passing through the whole atmosphere to the 



dr^ 



earth's surface, we may reckon ^ as unit; thus we get 



dA0=^^'rdn; 

 and by integrating 



5 ^ -. ^ ^ (P'^ r radn 



This result, which M. Biot has also obtained, is equivalent to the geometrical con- 

 struction communicated by Newton to Flamsteed in a letter from Cambridge, De- 

 cember 20, 1694. The problem was now reduced to the quadrature of a curve, for 

 which a general method is given in the fifth lemma of the third book of the Prin- 

 cipia, a method which is still used when the direct process of integration fails, or be- 

 comes too intricate for practice. What has been said not only proves the exactness 

 of Newton's solution of the problem ; it also points out, with little uncertainty, the 

 manner in which he obtained it. Of the arithmetical operations of the quadrature 

 there is no account ; and they would be of no interest had they been preserved. He 

 complains much of the great labour of the numerical calculations ; but all difficulties 

 were overcome, as was to be expected : a table was computed and communicated to 

 Flamsteed in a triple form, for summer, winter, and the intermediate seasons of spring 

 and autumn. On mature reflection there occurred to him a serious objection to the 

 supposed scale of densities, on which account he writes to Flamsteed that he does 

 not intend to publish the tables. The fault lies in this, that the centripetal force 

 which continually inflects the light to the earth's centre, is the same at all the points 

 of the trajectory, or, in the words of Newton, the refractive power of the atmosphere 

 is as great at the top as at the bottom, — than which nothing certainly can be more 

 diflferent from what actually takes place in nature. 



2 a2 



