Friction affects the Motions of Time -Keepers. 141 



Or, 



A.x4 ^ A 



Whence, if we denote by and i^A„ each of these inte- 

 grals, we have 



(A — A ) Z — -J, A — ^ A And 



n r V COS <p * 



. A- A 



r — T V sec <p 



^A— ^A 



n 



A table, then, of the values of ^A would enable us at once to 

 make the proper reductions. 



Integrating by means of the preceding formula, we find 



^A{ 1+ (I)%(H)V(^)' + etc} 

 -!-A( S 4) 2 {(^) 3 + (|||) 5 + e t e} 



- (H) si » A ( sin ^ ) 4 {( S ) 3 



+ etc . 



a formula remarkable for its symmetry, but unfortunately of no 

 use in practice, since the coefficients are all infinite. 



But if the integration be performed by means of the formula 



sinZ 2Z = cosZ -J— --sinZ 4- — -— sin Z T +_ r __L sm % ^ +etc I 



the value becomes, after several reductions, 



^A = A + sinA(si„|) 2 |(I) 2 { 1 } 



+ etc 



