484 SCIENCE PROGRESS 



other. The curve is then something of the shape of a horse- 

 shoe, which breaks up, for a still smaller value of C, into two 

 elongated pieces. These elongated pieces shrink quickly and 

 contract into two points, when C attains the minimum value (33). 



Thus there are four critical stages in the development of the 

 curves when C has the values between 40" 18 and 33. The first 

 stage is when the internal ovals unite into a figure of eight. 

 This evidently must occur between S and / and the point is one 

 of zero effective force where the planet may revolve without 

 motion relatively to the two other bodies (fig. i). It is found 

 that for the values assumed for v (10), here r=i—p and turns 

 out 717, p = "282 and C = 40" 18. Thus this point is the beginning 

 of a family of orbits. 



The second position, where the smaller end of the " hour- 

 glass " figure unites with the outer curve, occurs at a point in 5/ 

 produced beyond/, and here r = 1 -]- p. For this case the values 

 of p = -347, r'= r347, C= 38*88 (fig. i). This point lies beyond 

 Jove, on the line S J produced, and is similarly a position at 

 which the planet may revolve without relative motion, but, as 

 before, the motion is unstable. 



The third critical position, where the "horse-shoe" breaks, 

 is similarly in the line S /, produced beyond S, and here 

 p = r + I, The values for the quantities r, p and C are approxi- 

 mately r = 0*95, p — I "95 and C= 34*91. This is also a case of 

 unstable motion. The fourth position occurs when C is a 

 minimum, and the elongated pieces shrink to points. Here 



— = 0^ = giving r = I ; p = I ; C becomes 31/ + 3 = 33. If 

 pr op 



we draw an equilateral triangle on 5 / as base, its vertex will 

 be at this fourth critical point, and since two such triangles may 

 be drawn, one on each side of the base SJ, there are two points 

 corresponding to this position. 



This is a solution of the problem of three bodies, known to 

 Lagrange, the three bodies. Sun, Jove and planet standing at 

 each corner of an equilateral triangle, the latter revolving with 

 uniform angular velocity. It is approximately realised in the 

 actual solar system by the Sun, Jupiter and a minor planet 

 recently discovered, which revolves at a distance from the Sun 

 nearly the same as that of Jupiter. Though for the case of 

 V = 10 the motion is unstable, yet for values greater than twenty- 

 five it is stable. The five critical positions thus correspond with 



