MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 177 



at A. Now Bp is the space through which the centripetal force ^4^ would 



cause a molecule of light to move from a state of rest in the time dr: wherefore 



also 



2Bp = -'-^.dr^., 



T» B^ dz* dz ^ 



^P — sinBAD " 2R * Tdn' 



and, by equating the equal quantities, we get 



dss d.fiq) dr^ J , ^ 



The refraction of the light in moving from A to B, or the difference of the directions 

 of the curve at A and B, is evidently equal to the angle subtended by A B at the 



dz 

 centre of the circle of curvature, that is, to -n^ : wherefore if S ^ represent the refrac- 

 tion increasing from the top of the atmosphere to the earth's surface, we shall have 



_ _ ^•'P(g) . 



d 



dr dsz^ 



This formula is merely an application of the 6th proposition of the first book of the 

 Principia. 



Another general and useful expression of the differential of the refraction is easily 

 obtained. Draw C H = ?/, perpendicular to the tangent A H : from the known pro- 

 perties of curve-lines, we have 



but in this formula I must be conceived to increase from the surface of the earth to 

 the top of the atmosphere. 



In applying the last formula it is necessary to have a value of ^. Draw O L to 

 touch the curve at O, and C N perpendicular to O L : put §', v' for the density of the 

 air, and the velocity of the light at O ; also y for the perpendicular C N, « for C O 

 the radius of the earth, and for angle CON, which is the apparent zenith-distance 

 of the star : we shall have 



Area A B C <i ;^ 



T? = ^X2/ = «X2/: 



and because the curve is described by a centripetal force tending to C, the value of 



MDCCCXXXVIII. 2 A 



