176 



MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 



wherefore, upon the whole, the real increment of \P- will be 2 (p (^ -|- c? ^) — 2 <p (f) ; 

 so that we shall have 



and, by integrating, 



rf. t;2 = 2 ^ fe 4- <^f) - 2 ^ (^) = 2 rf. (? (^) ; 



t;2 = ,i2 + 2 ^ fe). 



It is obvious that r? 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, 



t>2 = 1 + 2 ^ (f). 

 Resuming the equation 



we have 



vdv = d.(p{§) = -^-j^ .dr; 



from which we learn, that the same addition which v^ receives by the refractive power 

 of the air at A, it will acquire by the accumulated action of the force ' f ^ at all 

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

 ' _^^ urging the light towards the earth's centre at all the points of the curve A B. 

 Thus the path of the light of a star in its passage through the atmosphere is a trajec- 

 tory described by the action of the centripetal force '-^j-^ tending to the centre 



of the earth, the sign — being necessary, because the analytical expression is essen- 

 tially negative. 



Draw A H a tangent of the curve at A, B m perpendicular to A H, and produce 

 C B to meet AU'm p: put n for the angle A C O which the radius vector A C 

 makes with C O the vertical of the observer; d z for A B the element of the 

 curve ; dr for the time of moving through AB ; and R for the radius of curvature 



