In the direction a: a': a" with speed: \/ 



Form corresponding products of (4) 



'(dxdydz) 

 and | , ■-. —, , " • , --- | add and integrate. 



(6) 



H 



[ d x 



dt 



I dt 



■1. 



J 



M 



(a'-+a/ 2 H a" J ) :im, m 2 ). 



In the direction '/., 

 with speed: T « : M. 



Multiply (4) by /° 

 and scalar-integrate : 



x ,df__M M 



2 T d t r j 2 a 



This is the equation of energy. It shows that the speed of a 

 planet increases when its distance from the sun decreases, and 

 ■eke versa. Also, since M=m! + m 2 is sensibly the same for all 

 planets, therefore the speed of a planet depends only on its dis- 

 tance from the sun and a constant. 2 a. of its orbit (later shown 

 to be its major axis). 



Forming corresponding determinants 

 of (4) with fx. y, z) and integrating: 

 x y z 

 d x d v d z i i i \ 



■' ,itd t . -dt- = ■••''.'■■w 



where P ];-[ lv -1 and c is positive. 



Multiplying corresponding terms by 

 I x. y, z), and adding, we find: 



f 1 x-| lj y-j I, z- •» ; similarly, 



(8) i if i, •;>■ i.'' z ' = i.. 



t d t 'dt dt 



Multiplying (4) by 

 /° and integrating the 

 vector part : 



where i c / =c. 



Taking the scalar pro- 

 duct by r we find 

 S i- f 0; similarly 



dt 



0. 



Equation ( 7 ) shows the rate of description of double areas by 

 the radius vector from sun to planet to be constant ( = c) and 

 that its motion is in a plane perpendicular to ( 1:L.:1 2 ) =*. The 

 direction of this axis is such that an ordinary screw, when made 

 to advance along it, will rotate- in the direction of the description 

 of areas. 



