222 NEWTON S PRINCIPIA. 



of the rest on these being each in the above ratio, the 

 resultants will also be in the same ratio, and the attracting 

 particles at the beginning of the motion being similarly 

 placed, and the forces in each system proportional, the 

 directions of the resultants will be parallel. Now we know 

 that two similar particles beginning to move in parallel 

 directions will describe similar orbits in proportional times, 

 when at the end of those times the directions of the forces 

 are parallel and proportional to the squares of the velo 

 cities and the reciprocals of any homologous sides of their 

 orbits. Hence these two particles begin to move similarly 

 under the action of such forces as tend to preserve the 

 similarity of their motions. And the same is true for all 

 homologous particles in the two systems. Hence all the 

 particles of the one system at the end of any small time T, 

 are placed similar to those of the other at the end of the 

 small proportional time T and are moving in a similar 

 manner. Hence the same thing will again be true at the 

 end of the next proportional intervals, that is, at the end 

 of the proportional times 2 T and 2 T . Therefore the 

 particles will continue always to move among themselves 

 with like motions and in proportional times. 



Let there be two fluids or systems such that the 

 particles of the one are similar to those of the other ; let 

 the diameters and distances of any two particles in one 

 system be d times the diameters and distances of the cor 

 responding particles in the other, and let the density of 

 these particles in one system be p times that of the cor 

 responding ones in the other system. Let the particles 

 begin to move from similar positions, and if we suppose 

 the forces in the two systems to be always proportional to 

 the squares of the velocities directly and the diameters of 

 the corresponding particles inversely, the several particles 

 will describe similar orbits in similar times. Let the velo- 



