NEWTON S PRINCIPIA. 223 



city of any particle in one system be v times that of the 

 corresponding particle in the other when at the correspond 

 ing part of its orbit. Let two large bodies which are 

 similar to each other in the same manner that two cor 

 responding particles are similar, be similarly projected 

 into these two systems. They will then describe similar 

 orbits in proportional times. The diameter of one body is d 

 times that of the other, and the velocity of one will be v times 

 that of the other. Let us consider the resistances to these 

 bodies : it will arise partly from the centripetal forces with 

 which the particles and the body act on each other, and 

 partly from the collisions and reflexions of the particles 

 and the body. The resistances of the first kind are, by 

 hypothesis, as the squares of the velocities directly and the 

 diameters of the corresponding particles inversely, and the 

 masses of those particles directly, that is, the ratio of the 

 resistances in the two systems is 



The resistances of the second kind are as the number of 

 reflexions and the forces of those reflexions. The number 

 of the reflexions in the two systems are as the velocities of 

 the corresponding particles directly, and the spaces between 



their reflexions inversely, hence the ratio is -&amp;gt; The forces 



of the two systems are as the velocities and masses of the 

 corresponding particles, hence their ratio is v. d 3 . p ; hence 

 the ratio of the resistances is 



-xv d^p = v* . d 2 . p 



joining these two ratios, the ratio of the whole resistance in 

 the two systems will be 



v 2 . d\ p. 



In such fluids, and under such conditions as those we 



