104 Trans. Acad. Sci. of St. Louis. 



This formula is exact and has a finite form, n x and n 2 being 

 interpreted as in § 7. 



The efficiency of Involute Teeth is as before 



where R is given in (16). 



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

 ing to the logarithmic series: 



nc or x 



log (1 + x) = x ^ + -3- — X +&C * 



Hence 



*! — «/,_/• L *f 



^ l0g ( 1+ _^) = 





r^ + wf 



±^ 1 o g (i + _^L_) =s 



Substituting these values in (16) we get 



R ~\n l + TJ 2 [n? 



11 



2 ni 7Tf 71? + 7T 2 / 2 



+ 



2/2 3 (ni 2_ 7r 2 / 2 ) 2 



(17). 



TT 2 / 2 71* + 3n 2 7T 2 / 2 



J — &c 



2 (M, 2 — 7T 2 / 2 ) 3 



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

 ing the divisions indicated by the fractions in the brackets. 



n? — tt 2 / 2 n* n* 



