8;.] PRINCIPLE OF AREAS. 47 



(9) Two spheres, whose masses are as i to 2, attract a particle of 

 mass i according to Newton's law ; the distance of the centres of the 

 spheres is = 4. Construct the equipotential lines in a plane passing 

 through the centres, by first constructing the equipotential lines for each 

 sphere separately, and then joining the points of intersection' whose 

 potential is the same. 



(10) A particle of mass m is subject to the force of gravity and 

 to the actions of two fixed centres C\, C 2 , one attracting with a force 

 inversely proportional to the square of the distance, the other repelling 

 with a force directly proportional to the distance. Find the equipoten- 

 tial surfaces. 



87. The Principle of Angular Momentum or of Areas. Let us 



begin with the case of plane motion, the equations of motion 

 being 



d' 2 



If we combine these equations by multiplying the former by y, 

 the latter by x, and subtracting the former from the latter, we 

 find 



- mx d ^-my d ^=xY-yX. (14) 



The right-hand member is the moment (with respect to the 

 origin) of the resultant force F whose components are X, F(see 

 Part II., Art. 91), while the left-hand member is an exact deriva- 

 tive, viz. the derivative with respect to the time of 

 mxdy/dt mydx/dt, 



as is easily verified by differentiating this quantity. The result 

 can therefore be written in the form 



and gives, if multiplied by dt and integrated, 



(16) 



dt dt 



These equations express the principle of angular momentum, or 

 of areas, for plane motion. 





