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



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

 interpreted as in § 7. 



The efficiency of Involute Teeth is as before 



E = 1 — R 

 where B, is given in (16). 



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

 ing to the logarithmic series: 



X 2 X 3 x i 



log (1 + aj) = x — -y + -y — -j + &c. 

 Hence 



n, — irf . ( . . 77-/ \ 

 — ^ log H ^— > = 



^ 2 / , *- 3 / 2 &c 



^1 2n i ( ?i ! — ^Z) 3?l l ( n i — ""/ ) 2 



'' / toK^l + 



V W, + 7T/ / 



"l/ °l «> + T/ 



7T 7T 2 / 7T 3 / 2 



„ 2 Wl (n 1 +7r/)^3n 1 (n 1 +7r/) 2 



w 



Substituting these values in (16) we get 



*-(-+-i£ 





+ 



n * — w *p 3 (n^—Tr 2 / 2 ) 2 



(17). 



7T 2 / 2 n* + dnfir 2 /* 



2 (n 2 — 7t 2 / 2 ) 3 



— &c. 



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

 ing the divisions indicated by the fractions in the brackets. 



Thus v =1 + ^y 2 , <fi + &c 



