Frictim affects the Motiom of Time-Keepers. 1 41 



Or, 



i(wj^=/{..(i)'K)%(t5)'(4)*.*}>-1 



• -/{.-(i)*(4)'-(S)'('4y-'}>i ■ 



Whence, if we denote by ^4- A and i^-An each of these inte- 

 grals, we have 



T 



(A — A ) — ==^ = >!' A — ^;' A And 



" T V cos ip 



A— A._ 



T =r T \^ sec (p ■ 



A table, then, of the values of 4'A would enable us at once to 

 make the proper reductions. 



Integrating by means of the preceding formula, we find 



.*.A{..(i)%(i-:)'.(S)'-"} 



--A{(i)%(H)^(S)'-"} 



+ 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 



/( I n+l 1 "+2 . ^" + 3, 1 n4-2 71+4 . _n + S, ^ ■) 

 • y" x 7 rn« 7 ■! _- - sin Z H sin Z + — -- — -- — — sin Z +etc v 

 sinZ JZ=cos/5 |— j^sm/r/ ^„-n „+3 n+l n+3 «+5 j 



the value becomes, after several reductions, 

 v^A = A + sinA(sin-) 3 ( g) | * / 



— (-^y-KaD'i-Q'-G-:)"} 



+ sin 



+ etc 



