64 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 51 



The relative motion will be referred to the coordinates (£, rj), moving 

 with the rotating system and which at the instant t = o (as the 

 figure shows) coincide with x, y. Denote by B the angular distance 

 from the y-axis at the time t in the case of the absolute motion, and 

 by /? the corresponding difference from the jj-axis in the case of 

 relative motion, evidently then 



p = B - cot (5) 



Further if r denotes the radius vector at the time /: 



(6) 



(7) 



From (5) Ave derive 



r sin 8 = r sin B cos cot — r cos B sin cot ) 



.... (8) 

 r cos /? = r cos B cos cot + r sin B sin cot J 



By substituting x, y from (2) in (6) ; r sin B and r cos B from (6) 

 in (8) ; and finally r sin /? and r cos /? from (8) in (7) ; we get 



? = (a — b) sin cot cos cot 



f) = b cos 2 cot -{- a sin 2 w2 = a — (a — 6) cos 2 cot. 



But for this can be written, by application of known goniometrical 

 formulae: 



f = sin 2 arf, 



cos2 cot -{ 



2 2 



(0) 



J (a — b) is the radius of the desired circle; and £(a + b) is the dis- 

 tance of its center m from the center of rotation M on the >;-axis. 



Instead now of starting with the construction of the absolute 

 motion, as here done, we can also follow the reverse process, assum- 

 ing the arbitrary relative velocity v in an arbitrary direction as 

 being given for any distance b of the body from the center of revo- 

 lution M. For simplicity it will however be for the present assumed 



