Woodward — The Efficiency of Gearing under Friction. 97 



Now the friction actually overcome in either case isP sin <p, 

 and since the velocity of sliding is r {a 1 + « 2 )» the amount of 

 friction overcome, or the energy lost, during the time dt is 



dU = Psin <p (« 2 + « 2 ) rdl (3). 



These formulae hold at all times and for all kinds of teeth 

 that are of correct outlines. 



3. Epicycloidal leeth. Let the driving moment M 1 be 

 constant, and let the teeth be epicycloidal, described by a 

 rolling circle with radius r . Also let q be the " arc of ap- 

 proach " i. e. the arc of one of the pitch circles which will 

 pass the pitch point while the point of contact T is moving 

 to the pitch point. 



It is evident that during the approach dq is negative, and 



that 



ajrfit = a 2 r 2 dt = — dq; 



hence 



{ a 1 + a 2 )dt = -(±+l.}dq. 



But from the figure, 



r = 2r cos d, and q = r (tt — 26), 

 hence 



(4) 



