THE PROBLEM OF THREE BODIES 



485 



particular exact solutions, three of which are unstable and two 

 are in general stable. The critical curves of the family 2 fl = C 

 are given in our figure, being symmetrical on both sides of the line 

 SJ, after Sir George Darwin's paper. It may be seen from this 

 figure that for values of C greater than 40' 18 the third body must 

 be either a superior planet moving outside the larger oval, or an 

 inferior planet inside the larger internal oval, or a satellite within 

 the smaller oval, but cannot exchange one of these parts for 

 either of the others. When Clies between 40'i8 and 38'88 the 

 body may be a superior planet, or an inferior planet, or a satellite, 

 or a body moving in an orbit partaking of the two latter charac- 



Curves of Zero Velocity (Darwin). 



Critical values : C = 4o'i8, C = 38'88, C — 34'9i, C = 33. 



^o(r^ + ^)+(,' + L)^C. 



teristics, but it cannot pass from the first condition to either of 

 the two latter. If Cis less than 38*88 and greater than 34'9i, 

 the body may move anywhere except within the horse-shoe 

 shaped region (shown in the figure). Here it is possible for a 

 body once started as a superior or inferior planet or satellite to 

 change the characteristics of its motion from one of these forms 

 into either of the others. When Cis less than 34'9i and greater 

 than 33, the " forbidden regions " consist of two strangely shaped 

 spaces on each side of the line joining Sun and Jove. For values 

 of C less than 33 the body may move anywhere. Since it is 

 found that for values of C greater than 40' 18 the third body must 

 be either a superior planet or an inferior planet, or a satellite, 

 and these cases are treated of in the Lunar and Planetary 



