Researches in the Theory and Calculus of Operations. 239 



The centrifugal force is geometrically deduced as follows : 



(Fig. 32) the angle At A', being in a semicircle, is a right 



angle, and gives the proportion AA' = d : At:: At: Ab; 

 (At) 2 



whence Ab — . A2 is *the measure of the original 

 tangential velocity impressed upon the body at the point 

 A. The whole force represented by the diameter d, when 

 measured by the effect A6 which it produces in the time 

 measured by A2, is really measured by the radius OA, 

 from which centre it proceeds, and acts as a constant 

 force (a force of the second degree) towards that centre, 

 upon the body at A (see section 3) ; where it enters into 

 composition with the tangential force in A2, and the re- 

 sult is the deflection of the course of the mobile into the 

 diagonal (or arc) At, with a velocity the square of which 

 x 2 (At)* 



is expressed by — = and which balances the force in 



O (Hutton). 



The conversion of circular into elliptic motion may be 

 traced by comparing figs. 34 and 35. In the first fig. 34, the 

 radius OA is an inextensible chord or line of balanced force. 

 The tangential velocity applied at A is compelled Jo follow 

 the diagonal AC by the unyielding force from the centre ; 

 for a similar reason, the course of the mobile is changed 

 from CB' to CC, and so on around the circle. But in fig. 

 35, the radius OA is a constant force of the second degree, 

 and generates a velocity AD which compounds with that 

 of the tangential force AB, resulting in the diagonal AC 

 the same as in the circle of 34. As the diagonal AC is greater 

 than the side AB of the parallelogram ABCD, the tan- 

 gential velocity CB r =AC is greater than AB, but, when 

 compounded with the constantly generated central velocity 

 CD=AD, gives the new diagonal CC'<AC. For, 0' is the 

 centre of the circle circumscribed to the ellipse of which 

 A A' is the major axis and the nearer focus. The diagonal 

 AC is obtained by compounding the original velocity AB 

 with the fixed tensional force O'A from the centre, and is 



