196 SEARLE. 



vary proportionally, while m'^= ?( + 3 = 1. If w>l, dR is affected 

 by a force greater than that affecting dY. While R is decreasing from 

 the value P, Y is increasing toward the ^■alue Q, so that the ratio 

 dR/dY is negative. The variation tending to change it is 7i + 3 times 

 as great as that which exists when «' = 1 , so that this variation is pro- 

 portional to {n -j- 3) dR/d Y. It may be desirable, however, to examine 

 this result more minutely, since it is to be the basis of subsequent 

 inquiry. 



In discussing the effect of forces, we are obliged to consider them as 

 applied, not continuously, but at intervals which we may ultimately 

 regard as infinitesimal. During each of these intervals, the rates of 

 variation, both of R and of }', are assumed to be uniform; that is, t 

 denoting time, dR/dt and dY/dt are constants. In the case of R, if no 

 actual force exists, the outward and inward forces are equal. In the 

 absence of force, the A-ariation of Y is uniform by the law of inertia. 

 In differentiating any function of R or of }', we accordingly regard 

 these quantities, for the moment, as independent ^•ariables. 



Let any particular value of R be denoted by Ro, and regard it as 

 constant for the moment. In the succeeding interval of time, we will 

 suppose i?o to be reduced to a smaller ^•alue R, so near Rq that the 

 difference is infinitesimal. To determine the ^•ariation of the force 

 acting on dR when R = Ro, we differentiate the expression m (i?o"'^^— 

 i?"+^)/i?o^ with respect to R, regarding Ro as constant. The resulting 

 differential coefficient is — m{n + 3}R"+^/Ro^, and disregarding the 

 difference between Ro and R, it becom.es — m{?i + 3) /?"~^ If we 

 regard the denominator as R^ instead of R^o, we have for the dif- 

 ferential itself - m[RH7i + 3)R''+--\- SRHRo""^'- i?""^)] dR/R^ The 

 second term of the numerator is of a higher order of infinitesimals 

 than the first, {Ro^'^— jR"+^) being infinitesimal. The differential coef- 

 ficient becomes as before — )n{n + 3)/?"~^ 



The variation of Y is not aft'ected by any outward force, but is 

 affected by the inward force on R in the ratio Y/R, as appears from 

 consideration of the right triangle shown in Fig, 2, of which R is the 

 hypotenuse while U and ]' are the legs. The effect of inward force on 

 R being - mR", its effect on Y is - mR"Y/Ro = ■- ?»i?o"-^ Y for 

 any particular value of R denoted by Ro, which is to be regarded as 

 constant for the moment. The differential coefficient is — mRo^~^, 

 and we have (n + 3) dR/dY for the ratio of the variations of force 

 at any moment affecting dR and d Y. By the definition of Z, d Y = dZ, 

 and the action of force upon Z is the same as its action on Y. The 

 ratio of corresponding variations of force upon dR and dZ is therefore 



