94-] PRINCIPLE OF AREAS. 5! 



93, The most important case of the application of the equa- 

 tions (21) arises when one or more of the conditions 



yZ-zY=o, zX-xZ=v, xY-yX=o ,(22) 



are fulfilled. The first of these conditions means that the pro- 

 jection of the resultant force on the jAsr-plane passes always 

 through a fixed point, viz. the origin of co-ordinates ; or what 

 amounts to the same thing, that the resultant force always 

 intersects the axis of x. Similarly, the second condition means 

 that the resultant intersects the axis of y. Hence, if both these 

 conditions are fulfilled, the resultant force passes constantly 

 through a fixed point, the origin of co-ordinates. 



It follows that if any two of the conditions (22) are fulfilled, 

 the third must also be fulfilled. This is also evident ana- 

 lytically, as any one of the three equations can be derived from 

 the two others. 



94, If the conditions (22) are fulfilled, the integration of the 

 equations (21) gives 



dz dy 



= 



dy dx 



dr my Tt =c 



where c v c^ C Q are constants depending on the initial condi- 

 tions. These equations express the proposition that if the 

 resultant force passes constantly through a fixed point, the angular 

 momentum about any axis passing through this point remains 

 constant. 



Multiplying the equations (23) respectively by x, y, z and 

 adding, we find 



> ( 2 4) 



which is the equation of a plane passing through the origin. 

 As the co-ordinates x, y, z of the moving particle fulfil this 



