100 Trans. Acad. Sci. of St. Louis. 



in which the action of only one pair at a time is assumed, and 

 from some other assumed relation between them. If two pairs 

 of teeth are always in action, the actual numbers of teeth on the 

 wheels must be divided by 2 in order to get the n i and n 2 of 

 the formula. If three pairs are always in action, we must 

 divide by 3. Thus suppose the actual numbers of teeth in a 

 pair of gears are 36 and 90, and that three pairs are constantly 

 in action ; then ^ =12 n 2 — 30.* 



8. Reduction of the value of R to a more convenient form. 

 If, as is usual, the arc of approach is made equal to the arc of 

 recess, we shall have 



7r irr 

 6=6 = J 



1 2 2 n t 2r 



2r 

 Now for convenience let the quantity — ° = e. This quantity 



appears in the values of 7c, 7c', 6 1 and 6 2 . 



Then 6=^-~ e ; k=(l-e)f;k' = (l+e)f. (11). 



Substituting this value of 6 for 6 X and 6 2 in (8), and e for its 

 value in the co-efficient of the same equation we have, 



R 



\n 1 nj 



?ijYy 



2tt 



— log ( cos — — & sin — ) — 7c — 



V n x e n Y e) n x e 



Jc' ( — ) — log ( cos — + 7c' sin — 



V?*^/ ° V n x e n x e 



1 + li*~ 



* The case in which two pairs are in action a part of the time and 

 one pair a part is not provided for by my formula. For such a case the 

 "approach" would consist of two epochs during the first of which M\ 

 would be double its value in the second epoch. The resulting value of B 

 would thus contain four definite integrals instead of two. I have not thought 

 it worth while to elaborate the result. 



