439 



means of one or the other of the two verbal formulae, in the other 

 three quadrants, and thus every particle (dm) of the disc contributes 

 to the same effect. This effect is due to the difference of the velocities 



-^- and at P and P f , or to the momentum ( -- | dm lost or 

 dt dt \dt dt) 



gained by the particle dm in the time dt. 

 The value of -^ is obtained from the equation y =r<j> sin 0, making 



both d> and to vary ; but the value of -%- is obtained from that of 



dt 



-- by making only to vary. It is thus shown that 



-? -- \dm=( cos . -i <i) . sin0 | 



dt dt) \ dt ) 



It is thence shown, by taking the moments about AB, and ap- 

 plying D'Alembert's principle, that 



-w * /s 



shi . cos 8dO= 



the integrals applying to only, and between the limits O and 2?r ; 

 i. e. to all the particles of the disc simultaneously and independently 

 of <j> or t. From this is obtained the result 



W being the weight of the disc. 



This value being periodical, and ranging between the limits O and 



the maximum - , shows that the disc makes oscillations which are 

 WrV 



of less extent and duration, as the spuming of the disc is more 



pi 

 rapid ; i. e. as ur is made greater compared with ; and thus if F 



denotes a small weight (such as is usually supplied with the appa- 

 ratus by the makers), the extent of the oscillation becomes insen- 

 sible. This formula, applied to the apparatus with which the ex- 

 periments were made, gives the theoretical maximum of about 

 18 minutes of a degree. It is evident that when F represents a 

 weight, it should be replaced in the differential equation by F cos <{>, 

 but the result practically coincides with that actually obtained when 

 F is not excessive. 



