OF DEFLECTIVE FORCES. 19,5 



(d^x" -{-d-ij- +d"Z'). But ds being constant, we Lave 



-2, ^(d=.^ +d^y^ +dcz^) =1 (271); and(g)= + (|)a 



/SV\ u- 



+ ^^j- = l (254, Sch. 2): consequently A =—, as has 



already been inferred, with respect to the central force in 

 a circle, from a simpler mode of Reasoning (258); but the 

 Coincidence is of use in strengthening the basis of the 

 analytical investigation. 



Now, if the surface be spherical, the curve described 

 will obviously be a great circle of the sphere, and its radius 

 of curvature that of the sphere, since the deflection can 

 only be in the direction of the radius, and in the plane in 

 which the body moves. And if a thread be substituted 

 for a surface, the tension of the thread will be equivalent 

 and equal to the pressure on the surface. 



The whole pressure on the surface will be obtained, by 

 adding to the centrifugal force any extraneous forces which 

 may be acting on the body. And since the force always 

 acts in the direction of the plane of the body's motion, 

 when that plane is not perpendicular to the surface, the 

 pressure on the surface will obviously be reduced in the 

 proportion of the radius to the sine of the inclination of 

 the plane to the tangent plane ; the remaining portion act- 

 ing in the direction of the surface, and. requiring to be 

 counteracted by some other force. But in the absence of 

 such forces, it has been shown that the centrifugal force 

 is simply equal to the pressure on the surface ; the plane of 

 the motion is, therefore, in that case, always perpendicular 

 to the surface. 



