52 



Mr. G. J. Stoney on the Physical 



[Recess, 



stars would be reduced. If tlie resistance acted only at the apse of 

 the orbit, the diminution of the mean distance would be effected by a 

 shortening of the aphelion distance exclusively, the perihelion distance re- 

 maining unaltered. But since the resistance is not confined to this spot, 

 but acts also for some space on either side of it, the perihelion distance 

 will at each passage undergo a slight decrease, which would inevitably 

 cause the stars in the end to fall into one another, if the tangential resist- 

 ance were the only force disturbing the orbits. But there will be normal 

 forces also. The resistance to which each star is subjected in passing 

 through the atmosphere of the other is a force neither directed through 

 its centre, nor parallel to the tangent of its orbit, since an atmosphere is 

 not a thing of uniform density. Since these forces are not parallel to 

 the tangents of the orbits, they will produce normal components, which 

 will be directed outwards ; and since they are not directed through the 

 centres of the stars, they will cause the stars to rotate, and these motions 

 of rotation, which will take place in the same direction in which the stars 

 are revolving in their orbits, will in the subsequent perihelion passages cause 

 each star to sweep the atmosphere that opposes it downwards towards 

 the other star while bursting through it. It will accordingly itself suffer 

 an equal reaction, which will be another force normal to its orbit and 

 directed outwards. Such forces will lengthen the perihelion distance, while 

 they leave the mean distance undisturbed*. Accordingly the combined 



where /3 is the perihelion distance, and the other letters have then - usual significa- 

 tions. 



A tangential resistance acting at any point of the orbit diminishes v, and therefore In- 

 equation (1) diminishes a, the mean distance. 



To find its effect on /5, the perihelion distance, transform the second equation by 

 putting 



P>=p.{l-x); (2) 



whence, neglecting the higher powers of cc, since we only seek the effect of a resistance 

 acting in the neighbourhood of the perihelion where x is small, 



l-l =x (t-l\ ( 3) 



p r , \fi p / 



From equation (3) it appears that if v is diminished whiles and r continue unchanged, 

 w must increase, and therefore by equation (2) jG, or the perihelion distance, is reduced. 



* This appears from the foregoing equations by supposing p to receive an increment, 

 while v and r remain unchanged. Equation (1) is not disturbed; in other words, the 

 mean distance is unaffected. Equation (3) shows that x becomes less ; and equation (2) 

 that /3, or the perihelion distance, is increased both by the increase of p and the diminu- 

 tion of sc. The reverse effect upon /3 is produced by a decrease of p. Nowjt? is increased 

 by the normal forces from the time the stars touch up to the moment of the perihelion 

 passage, and decreased during the second half of the transit. Accordingly /3, the peri- 

 helion distance, is first increased and then diminished. If the stars behaved to one 

 another like perfectly elastic bodies, these changes would be equal, and would cancel one 

 another. But at each transit vis viva is converted into heat, in other words the stars 

 do not behave like perfectly elastic bodies, and the mechanical forces elicited during the 

 second half of the transit are feebler than those during the first. Hence there will on 

 the whole be an increase of the perihelion distance. 



