238 Researches in the Theory and Calculus of Operations, 



A direct deduction of the measure of centrifugal force 

 can be shown by beginning with fig. 36, in which AB re- 

 presents the path of uniform movement, Al = l 2 = 2 3 = 

 3 4, etc. successively in equal times. At the fixed origin O 

 is an orifice permitting the passage of an inextensible chord 

 * having an attachment at its farther extremity with the unit 

 mobile 1^. OA is the common altitude of the successive tri- 

 angles all erected upon equal bases : the areas of the triangles 

 are therefore equal, and are described in equal times under 

 the even traction of the chord as delivered with just suffi- 

 cient friction to preserve its rectitude. 



Begin the operation anew, by fixing the inextensible 

 chord at (fig. 34). The unit step of movement cannot now 

 lengthen the distance of the mobile from the origin 0, but 

 is compelled to take the diagonal A C of the parallelogram 

 ABCD. The direction of the movement must consequently 

 be continually changed, while continuing uniform in time. 

 The bases of the successively described triangles remain 

 equal, and the radius of the described circle is their common 

 altitude : consequently here also equal areas are described 

 in equal times. The force at the centre which controls this 

 movement is a constant force, a force of the second degree, 

 whose measure is expressed by x 2 l t the distance it would 

 carry the unit mobile 1^ in the time xl t (equally expressed by 

 x 2 l t , the square of the time, when 1, =1,). For movement 

 in the circumferences of the circles OA and OA' (fig. 33), 

 these distances will be x 2 l t and x ,2 l t . Now in uniform travel, 

 the greater the distance the longer time, and arcs of cir- 

 cles are as their radii; then in the circles OA and OA', 

 of radii r and r', there arises the proportions 



Force expended in arc as : Force expended in arc a's' : : 

 , Time in as : Time in a's f ; 



or, substituting the measures, 



xPlt : x n \i : : rl t : r'l,, which gives the equation 



expressing the centrifugal force (Newton). 



