Mr. Ivory on the Theory of the Astronomical Refractions. 343 



matical 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 de- 

 crease in ascending in the same proportion that the distances 

 from the earth's centre increase. Now a and r denoting the 

 sam e thingsas before, put I for the total height of the atmo- 

 sphere ; then <fi (p) the refractive power of the air at the di- 

 stance r from the centre of the earth, will, according to this 

 hypothesis, be expressed by the formula 



/ N , /. l — r-\-a 

 ^(p) = <^(?') X 1 > 



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 



dr- 1 1 



dn 



dz^ V- l+2<p{p)' 



and as 2 $ (p), or the increment of the square of the velocity 

 of the light is very minute, amounting to less than '0006 in 

 passing through the whole atmosphere to the earth's surface, 



dr^ . , 



we may reckon —j—^ as unit ; thus we get 



d.U = ^.rdn; 

 and by integrating 



- (1) J 2 



^ a ' 

 This result, which M. Biot has also obtained, is equivalent 

 to the geometrical construction communicated by Newton to 

 Flamsteed in a letter from Cambridge, December 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 Principia, a method which is still used when 

 the direct process of integration fails, or becomes too intricate 

 for jwactice. What has been said not only proves the exact- 

 ness 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 



