Products, Quotients, Logarithms, and P&wers of Numbers. 177 



with the axis of the frame EF by two bevel wheels, as that 

 the number of revolutions made by the latter shall bear a cer- 

 tain given ratio ?i' to the number made in the same time by 

 the former; then will the number N4 shown at any time by 

 the index of this last frame be the cube of the number N, shown 

 at the same time by the index R, 



For — ^ representing the number of revolutions made by 



2 



the first frame EF whilst the number Nj is registered, n' -^ 

 will represent the number of revolutions made in the same 

 time by the second screw, so that n' -^. \ will represent the 

 distance by which the second wheel will have been moved 

 from the apex of the cone, and «' — ^ \ sin 1 the radius of the 

 circle which its point of contact with the cone will be in the 

 act of describing. Moreover, 27r ?- is the ande which the 



cone describes whilst AN, is registered by the index R : so 



N AN 

 that n' — ^ X sin J .27r i is the small arc described during this 



"2 "1 



time by the point of contact of the second wheel with the cone. 

 But the increment AN4 being registered during that time by 

 the index of the second frame, the circumference of the second 



wheel will describe the arc ^.2wp {71^ representing the num- 

 ber of equal parts into which the circle is divided to which the 

 index of the second frame points) ; 



but N2=N,2 (by equation 4) ; 



^. /.^V^Yn^.an, 



Therefore, passing to the limit and integrating, 

 /«^.^Asin.\j^ 



If therefore n! be so taken that 

 ■'4 





Sn^n^ 



(8.) 



