TELEPHONE DIAPHRAGMS. 471 



the orbit of the particle will dwindle, until finally the particle will fall 

 into the center O. This means that the orbit, instead of being a circle, 

 will be an equiangular spiral, in which the tangent PP^ at any orbital 

 position P, makes an angle of less than 90° with the reversed radius 

 vector PO. The instantaneous acceleration will be directed along the 

 tangent QQ^ of the spiral at the point Q, (180° — y°) in advance of P. 

 The centripetal force will be directed along PO, the frictional force in 

 the direction P^P, or parallel to QO, and the inertia force in the direc- 

 tion Q^Q, or parallel to VO. These conditions are represented in the 

 instantaneous force diagram, Figure 31, where Op is the centripetal 

 force —sx, Oq is the frictional force —rx, and Ov is the inertia force 

 —mx, X being the instantaneous displacement: 



x= Xoe^-^+^"">^ cms Z (24) 



so that 



r.i- = r (— A + joo')xo 6^-^+-''"')* = r (— A + jo:')x dynes Z (25) 



and 



mic = m (- A + jw')^ Xo e(-^+^'"')« = m (- A + jo^'Y x dynes Z (26) 

 For equilibrium we require that ^° 



— sx — rx — mx = dynes Z (27) 



or 



-s -r(- A + jo:') - m (- A + jcoj = '-^^"^ Z (28) 



cm 



radians 



whence w' = Vajg^ — A- = coo sin 7 (29) 



where — = cos 7 numeric (30) 



coo 



Each of the quantities x, x, '.i, pursues an equiangular spiral around 

 the center 0, or may be considered to pursue a circular path with uni- 

 form angular velocity, subject to an independent damping factor e"^'. 

 In the case of simple rectilinear vibrations, the projections of the 

 spiral motion may be taken on a reference axis COD. The initial 

 value of the velocity .i'l must be such as meets the physical conditions 

 of the system at the moment of a sudden change in the impressed 

 vibromotive force. After the change, the vibratory motion will be 

 the sum of the two terms in (17) or, the sum of the projections of the 



20 Bibliography, No. 8. 



