Products, Quotients, Logarithms, ayid Powers of Numbers. 1 73 



both released, and the cone turned until the index R points 

 to any number, the index S will point to the square of that 

 number. 



Theory of the Instrument. 



Let J be taken to represent half the angle at the apex of the 



cone, p the radius of the wheel PQ, and X the distance between 



each two threads of the screw ; also let n■^., n^, 7/3 represent 



severally the numbers of equidistant divisions on the circles 



to which the indices R, S, T severally point; then will 



N N N. 



— -, —-, — ^ represent the numbers of revolutions made se- 



«j ;?2 7}^ ' 



verally by the indices whilst the numbers Nj, Ng, Ng are 



registered. Now suppose that when the wheel PQ is at the 



apex of the cone the indices R, S and T each point to zero, 



and the frame EF being fixed, let the screw CD be turned 



until the index T shows the number Ng ; then will the screw 



N 

 have made — - revolutions, and the wheel PQ will have ad- 



"3 

 vanced from the apex of the cone to a distance AP from it, 



N 

 represented by — ^ . X ; so that the radius PM of the circle de- 



scribed by that point whilst the cone revolves will be represented 



N N 



by — ^ X sin «, and its circumference by 27r — ^ X sin ». Now 



' "3 «3 



let the cone be made to revolve (the wheel PQ being fixed 



upon the frame and the frame released) until the index R shows 



N 

 the number N,, or until the cone has made — - revolutions : 



then will the point P in it have been made to revolve through 

 a space represented by 



^ N3 . N, 

 27r— ^.Xsmi. — -, 

 Wg 7?i 



, SttX sin < ,. T -vT 



or by — — — .N1N3. 



"l''3 



But by reason of the continual pressure of the cone upon the 

 edge of the wheel PQ, that wheel will have been carried round 

 in the revolution of the cone, so that the circumference of 

 the wheel will have described precisely the same space as the 

 point P of the surface of the cone ; if therefore Ng represent 



N 

 the number pointed to by the index S, so that — ^ represents 



"2 

 the number of revolutions made by the frame EF and wheel 



