and on the Theory of Escapements. 107 



Now since x = a . sm nt + b, 



dx —, — T , -T — r dh , . , , , da 



.-. ~r = na cos nt + b + a cos nt + o . -j- + sin w< + o . -r- • 

 at at at 



dx 



But the velocity or -n has been assumed = na cos nt + h\ con- 

 sequently, 



Y db . r da 



a cos M^ + . -7T + sin w< + . -rr = 0. 

 at at 



dx 



And since -n = na cos nt + 6; 



<i*a; „ . — : — r ■ r db — ; — r da 



.'. -rz: = -na sin nt + b — na sm nt + b . -r- +n cos n< + o. -jr. 

 ar fflf at 



d~ X 

 Substituting the values of x and -j^ in the original equation, 



we find 



■ , , da . db 



n cos nt + b.-j- — na sm nt + b . -J- —j. 



Combining this with the equation above, or 



da — ■ r db 



sin nt + b . -J- + a cos nt + b . -^ = o, 

 da f db _ f 



we find -J- ='L . cos nt + b, -r- = - ^— . sin nt + b. 

 dt n dt na 



If we could solve these two differential equations, we should have 

 the complete determination of the motion. 



In few cases, however, it is practicable to obtain an exact 

 solution: and in all an approximation is sufficient for our pur- 

 poses. This may be obtained by integrating the expressions 



f f 



— cos nt + b, and — - sin nt + b, 

 n ' na 



o 2 



