19,6 CELESTIAL MECHANICS. 1. ii. 10. 



The curve thus described, on a spheroid, has been called 

 the perpendicular to tlie meridian : and it traces, as has 

 already been observed, (266) the shortest distance between 

 any two places in its direction. It does not, however, 

 remain actually perpendicular to the meridians which it 

 crosses, but is conceived to be traced by levelling, in the 

 same way as a flattened wire would trace it when bent on 

 the spheroid. 



[Scholium. It follows from considering the propor- 

 tion of the sagitta of curvature in a perpendicular and in 

 an oblique plane, that the radius of curvature must always 

 vary in the direct ratio of the sine of the inclination of the 

 planes, so that the pressure on the plane is the same whe- 

 ther the body move in a great or a lesser circle, the imme- 

 diate centrifugal force being increased, by the increase of 

 curvature, in the same ratio that its action with regard to 

 the surface is diminished, provided that the velocity be the 

 same in both cases.] 



273. Theorem. If a body move in a 

 resisting medium, and be subject to a uniform 

 gravitation in a vertical direction, its motion 



will be defined by the equation — =-^2"t > ^ 



being the space described in the direction of 

 the motion, z the vertical ordinate, a: a horri- 

 zontal one, c the resistance, and g the force 

 of gravity, dx being constant : and if the 

 resistance is as the square of the velocity, and 



A =:— » TT= ^^ > ^ being a constant quan- 

 tity, and hle=l. 



