286 Mr. Ivory on the 'llicory of the Astronomical Rcfi actions, 



scribed by the action of the centripetal force ^^ tending 



to the centre of the earth, the sign — being necessary, be- 

 cause the analytical expression is essentially negative. 



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

 to A H, and produce C B to meet A H in p: put n for the 

 angle A C O which the radius vector A C makes with C O 

 the vertical of the observer; rfz for A B the element of the 

 curve: cLt for the time of moving through AB; and R for 

 the radius of curvature at A. Now Bp is the space through 



which the centripetal force — '^J would cause a mole- 

 cule of light to move from a state of rest in the time dri 

 wherefore 



also 



XX _ Bffl _ dz^ dz 

 '^^ ~~ sin B A D "" 2 R ' rdn' 



and, by equating the equal quantities, we get 



4A=_^^(£.). ^^.rdn (1.) 



R dr dz^ 



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 evi- 

 dendy equal to the angle subtended by A B at the centre of 



d z 

 the circle of curvature, that is, to -^ : wherefore if S 9 re- 

 present the refraction increasing from the top of the atmo- 

 sphere to the earth's surface, we shall have 



rf . 8 9 = J-^ . -— 3- .rdn. 



dr dz' 



This fornuda 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 = ^, perpen- 

 dicular to the tangent A H : from the known properties of 

 curve-lines, we have 



r dr 



R = 



wherefore 



dy 



dz _ , dz }_ _ _ ^3/__ 

 IT ~^'~d7 ' r ~ v^;tZp 



