124 CELESTIAL MECHANICS. I. ii. 9- 



readily understand that dx^ is generally used for {(\xy and 

 d^x\ A^x\ for (d-x)2, (A-^xf ; and not for d(a"), (\\xf, as 

 might have been done without impropriety, if it had been 

 equally convenient. 



272. Theorem. The pressure of a moving 

 body on any curve, derived from its centrifu- 

 gal force, is expressed by the square of the 

 velocity, divided by the radius of curvature : 

 and the pressure on any surface is expressed 

 by the square of the velocity divided by the 

 radius of curvature of its path, and multiplied 

 by the sine of the inclination of the plane of 

 the curvature to the plane of the surface. 

 (§. 9. p. 23.) 



The equation Vdu:=zXS^s (250) affords us here the con- 

 ditions of equilibrium between the forces depending on 

 the curvature, and the pressure; but those forces are 



— — , --^, and — — , (233, 264), or, since ds=:vdt, dtzz 

 di'^' dV' df^ ^ ^ 



v^' "^^ h^'^'h h^'^' t^'^'y* ^"^ h^''' ^^''^ "^"''^ 



be respectively equal to F ^^ , V -^^ , and F f^, or, in 



ex hy hz 



this case, putting A for the pressure, and r for the perpen- 

 dicular to the surface, to A --, A rr-, and A -— , since the 



ex hy cz 



forces in each direction must balance each other. We have 

 consequently, adding together the squares of each equa- 



^--1(&+(|>-©1=(£)Md^.=.dv 



i 



