104 



Trans. Acad. Sci. of St. Louis. 



This formula is exact and has a finite form, n^ and 7i^ being 

 interpreted as in § 7. 



The eflSciency of Involute Teeth is as before 



^' = i _ /? 



where H is given in (16). 



13. A more convenient form of (16) is obtained by resort- 

 ing to the logarithmic series: 



/y»* /y»3 rv*^ 



log(l+a;) = x— -y^-g- — ^ +&c. 

 Hence 



"t — '^/l.^/i , '^Z 



«l/ 



log(l+_^) = 



IT 



-^v 



3r2 



+ 



7r3/ 



2nj (71 J — irf) 371, {n^ — irf f 

 1^ lo<. /l + ^-^ ) = 



&c. 



^i + tt/ 



TT 



Try 



+ 



3^.2 



n, 271, (n, + tt/ ) 3?ij {7\ + tt/ )^ 

 Substituting these values in (16) we get 



&c. 



R 



\ n^ 7*2 / 



2 



2??j7r/* n,2 _|_ 7;.2y 



2y^2 



■y^ 



w—TT-^f) 



+ 



7r2/2 ,^4 ^ Zn^rj^if 



2 (,i^2_^2y2y 



&C. 



(17). 



14. Formula (17) may be still more reduced by perform- 

 ing the divisions indicated by the fractions in the brackets. 



Thus 



71,2 ^ , 7r2/'2 rj^ifi 

 \ . = 14- -^ I ^ U RrC 



