relative to Stationary Motions. 273 



Willi the aid of these expressions the equation 



can be transformed into the following : — - 



TteiJ5fiL(W«+rV) '. (21) 



Putting now r and in the place of the variables universally 

 denoted above by q x and q q , we obtain 



dT _ m/j, f 

 dr' m + fi 



Pi — 



22) 



1 ~ (10 m + fji 



From this results further, if R, R', B, 6' denote the initial 

 values of r, r\ 0, 0\ the equation 



K p8tf-h8k_ mp r'S^r-mR + rm^e-Wm® ,_ Q . 

 2, - _ — - . (23) 



For the definition of the phases cf> l and (/> 2 we have, according 

 to (12), the equations 



t = h<i>i = h^', (24) 



and the question now is, whether the time-intervals i x and i 9 

 can be so determined that the mean value of the expression given 

 in (23) shall vanish with increasing time. With even a super- 

 ficial consideration of the motion in question, it is seen at once 

 which time-intervals are to be selected as i { and i 2 , because the 

 motion can be analyzed into two constituents — the alternate 

 approach and recession of the two points, and the rotation of 

 their connecting line — which can be considered singly, as varia- 

 tions of the quantities r and 6. 



Thevariation of r is periodic; and if we take the duration of 

 its period as i v the portion of the fraction in (23) relating to r, 

 viz. 



t 



the numerator of which changes only periodically, evidently 

 fulfils the condition that its mean value vanishes with increasing 

 time. 



For the interval which refers to it will be convenient to take 

 the time of a rotation of the connecting line, therefore the time 

 in which the angle increases to 2tt. But as the successive 

 rotations do not generally take place in equal times, we will un- 



PhiL Mag. S. 4. Vol, 46. No. 306. Oct. 1873. U 



r'^r-R'SR 



