and on the Theory of Escapements. J 27 



let the product of the force by the space through which they 

 continue their action be G and K respectively. Here c, e, and k 

 depend on the construction of the clock, and z on the situation 

 of the pallet when the pendulum is at the middle of its vibration, 

 that is, on the position of the clock. Observing that when the 

 pendulum is going, we must take 



1 r _f^ 



and when it is returning. 



rt-ifitJ ^ 



f^ 



for the proportionate increase, and observing that the first and 

 third forces are positive, and the second negative, we find for the 

 proportionate increase 



1 fr/ r^^ — - r-i — ; — r-^\ G (z - e) K(z-k) ■» 



or approximately, 



^F.c.z+^ -G.J^+KJ^\, 



1 trn'a' 



This will be nothing when 



c 2Fc'' + 4Ge-4Kk c „ 



z = — - X —rTT, — -vi TT?- = a small quantity. 



2 2Fc- — 2Gc + 2Kc 2 ^ ^ 



It appears, therefore, that the time of action on the pallet before 

 the pendulum reaches the middle point must exceed, by a very 

 small quantity, that after it has passed the middle point, in order 

 that the time of vibration may be independent of the maintaining 

 force. If the action began before this time, there would be a term 



of the form ^ — , or ^ nearly, in the time of vibration ; 



that is, the clock's rate would be made quicker, but less for large 



