Mr. Ivory on the Theory of the Astronomical Refractions. 285 



density p : on the other hand, the refractive power of the stra- 

 tum will augment u' by the quantity 2 <p {p + d p) : where- 

 fore, upon the whole, the real increment of u^ will he 2 <p {p 

 + d p) — 2 ^ {p) ; so that we shall have 



d.v^- = 2<i>(p + dp)-2<p{p) =2d.<p{p); 

 and, by integrating, 



0^ = ?j^ + 2 <f) (p). 



It is obvious that n- is the square of the velocity of the 

 light before it arrives at the atmosphere, that is, when it 

 moves in a vacuum. We may consider n^ as the unit in parts 

 of which the squares of the velocity of the light at the several 

 points of the trajectory are estimated ; which requires that 

 the formula be thus written, 



_ 0-2 = 1 + 2 4) {p). 

 Resuming the equation 



d.u^ = 2d.(p{p), 

 we have 



udv = d.4>{p)='^-4^.dr; 

 ^'^ dr 



from which we learn, that the same addition which o^ receives 

 by the refractive power of the air at A, it will acquire by the 



accumulated action of the force 'T' ^^ at all the points of 

 the line d r, or, which is the same thing, by the action of the 



r d . ^ (p) 



torce — - — !- urging the light towards the earth's centre at all 



dr on o 



the points of the curve A B. Thus the path of the light of a 



star in its passage through the atmosphere is a trajectory de- 



