Prodttcts, Quotients, Logarithms, and Powers of Numbers. 175 



N 



— ^X; so that the radius PM of the circle in the act of being 



described by the point P upon the surface of the cone whilst 



the numbers AN, and ANg are being registered, will be 



N 



— ?Asin»; and the exceedingly small arc described by that 



"2 



point whilst these numbers are registered, will be represented 



by 



Stt . — -'' A sm f. 



"1 "2 



But this arc, described by P upon the cone, is equal to that 



described in the same time by a point in the circumference of 



AN 

 the wheel, which last is represented by 27r ? p ; 



... 2.M..^.Xsin.=2.^^.; 



... (^AN,= ^; 



or passing to the limit and integrating between the limits unity 

 and Ng, 



(ii|i)N.=,„g.N,; 



.-. (s «lf ) =. 



:N,. 



Let the quantities X, «, n^^, p be so assumed that 



Xsin ( 



8"^= 10; (/3.) 



.♦. lON.rrNa; 



••• Ni=log,oN2 (3.) 



whence it follows that the number shown by the index R 

 (under this adjustment of the instrument) will in every position 

 be the common logarithm of the number then shown by S. 



If the spindle AB be made to carry round with it the screw 

 CD by the intervention of two bevel wheels the number* of 

 whose teeth bear a given ratio, ?i, to one another, so that the 

 screw CD may make?nevolutions or parts of a revolution whilst 

 the spindle AB makes one; if, moreover, the wheel PQ be so 

 released from the female screw, which forms its centre, as that 

 the outer surface of that screw may serve for an axis about 

 which it may turn freely whilst it is still carried along the 

 frame by the longitudinal motion of the screw; and if the in- 



