k 



OF SYMBOLICAL GEOMETRY AND MECHANICS. 505 



This is the general symbolical expression for the force acting on the planet, and it consists 

 of three parts whose direction units are a, /3, 7, that is, which act, along the radius vector, 

 perpendicular to it, and perpendicular to the plane of the orbit. Hence, if P, Q, S be the 

 forces which act on the planet in these three directions respectively, we have, 



d'r 

 d? 



P= ^- '•W, 



dr d{r io-i) 1 d{r^w^) 



^ dt dt r dt ' 



S = r 0)30)1 J 



which are the general equations for determining the motion of the planet, and of the plane of the 

 orbit. 



37. To determine the motion of a particle acted on by a central force varying inversely as the 

 square of the distance. 



Using the same notation as in the preceding Article, it is clear that the symbol of the force is 



~~? ' 

 and therefore we have 



d?u /la 



dF=^--r- ^')- 



Performing the operation Du on each member of this equation, and observincr that 



Du.a = rDa.a = 0, we have 



^ d-ti 

 Du. - — = 0; 

 rib- 

 and therefore D u — = constant (2) ; 



dt ' 



du du 

 for the former equation is evidently the differential of the latter, observing that D — . — = 0*. 



Q/v at 



Now u = ra, — = — a + rwli (writing w instead of 0)3), and therefore, since Da -a - 0, and 

 dt dt 



Da . (i ^ y, the equation (2) becomes 



rcoy = constant. 



Hence, y is an invariable direction (i. e., the motion is in one plane) and r"w is constant, equal 

 to // suppose. 



d/3 

 Now -!- = - wa, and therefore (1) becomes 

 dt 



d-u n d/3 fi d/3 



dt' r-fo dt h dt 



It in obvious that iHDu.v) = Ddu .a+Du. dv. 

 •■j T 2 



