144 SEE— DYNAMICAL THEORY [April 19, 



density, and having a thickness R-r, it is evident from equation (7) 

 that 



V =^ r "" d4> r sin ddd f ardr, (46) 



and the forces along the coordinate axes will be 



X' = f cos (f)d(f) r sin^ OcW f adr, 



y = f sin <f)d(f> r sin' Ode f adr, (47) 



Jo Jo Jr 



Z' ^ { dcf) f cos e sin OdO f adr. 



Now every globular cluster may be regarded as made up of a series 

 of such shells; so that the total forces become 



X=Yi A". = Z cos c\>dci) sin- Odd adr, 



i = \ 1 = 1 Jo Jo Jr 



y=T,^i = 12 r " sin (^# psin' Ode f adr, (48) 



»=1 ! = 1 Jo Jo J I- 



Z=Y. = Z d(l) \ cos e sin ede adr. 



i=.\ i = l t/o t/o Jr 



These expressions are so complex, that we are obliged to re- 

 strict our consideration to the action of a single shell. Accordingly, 

 we shall suppose the single shell filled with stars to a considerable 

 density, and the distribution uniform. An external star coming in 

 from the distance, if otherwise undisturbed, will revolve in a Kep- 

 lerian ellipse having its focifs in the center of the shell. The mass 

 acting as if collected at the center is 4710-;'- (i?-r), where the thickness 

 R-r is not too large ; and the velocity accjuired at the outer border 

 of the shell is 



V^^k^i^-rrarXR-r)]^--'^, (49) 



where a is the semi-axis major of the Keplerian ellipse. 



