240 Researches in the Theory and Calculus of Operations. 



assumed the same as that produced by the velocity generated 

 (beginning at zero) by the constant force OA of the second 

 degree, during the same time. The effect of this force is there- 

 fore CD' greater than AD that in the circle, and conse- 

 quently inclines the terminal extremity of the resultant 

 diagonal towards the diameter A A'. But while continually 

 bending its further extremity towards the diameter, in virtue 

 of the attraction in the focus O, which varies with the 

 length and inclination of the radius, the diagonal departs 

 further and further from that focus, till finally arriving at 

 the diameter at the point A', where the intensional effect 

 of O, from increase of distance, is diminished to A'D, the 

 tangential tendency being A'b. From this point, then, the 

 diagonals continue to increase, and the right hand half of 

 the elliptic polygon repeats the left hand half in reverse 

 order. By actual geometrical admeasurement, it can be 

 shown that the areas of these several triangles are all equal, 

 and prove Kepler's principle of the description of equal 

 areas in equal times. When, in the unit of time, the primi- 

 tive impulsed or rotative velocity in A is less than the effect 

 pf the force in O, the resulting orbit is elliptical; 

 if equal to the effect " O, " " " parabolical: 



if greater than " " 0, hyperbolical (Laplace, Biot)« 

 To return to the globe represented in section in fig. 33, 

 the volume of emanated force at the distance r from the cen- 

 tre is proportional to r 3 the cube of the radius; at this dis- 

 tance, the intension of the force filling a spherical shell of 



unit thickness is measured by ^ the inverse cube of the ra- 

 dius. The emanated force filling a plane central section of 

 unit thickness is proportional to r 2 ; at this distance, a ring 



of this plane of unit thickness is measured by i the inverse - 



square of the radius, and this is the force that holds the 

 planets in their orbits. The primitive tangential or rotative 

 velocity depends upon the angular velocity, which, if counted 

 by equal distances in equal times, is proportional to the in- 



