TRANSACTIONS OF THE SECTIONS. 37 



described on the same supposition, in respect to other positions of the integrating 

 wheel or other values of the function /(^) between the hmits ^j and 6^, will be repre- 

 sented by the sum ^i 



2/(^.A^. 



If, therefore, the broken or interrupted variation here supposed to be given to the 

 distance of the point of contact from the centre of the plate (and which may be con- 

 ceived to be communicated by a jagged or step-like form of the edge of the rail) be 

 replaced by the continuous variation actually communicated to it by the curve, this 

 sum will pass into the definite integral* 



L 



'f{6).d6. 



Now if N represent the number of revolutions and decimal parts'of a revolution de- 

 scribed by the frame or integrating wheel, and ^ the radius of that wheel, the sum 

 (or whole space described by the circumference of the wheel round its axis) is re- 

 presented by the product 2 sr g N. So that 



2xgN= /* / 



Let now a contrivance be applied to the instrument for registering the revolutions 

 N of the frame, and therefore of the integrating wheel, to the 10,000th part, or to four 

 places of decimals. This registration may be made by the common method of as- 

 tronomical instruments ; or more conveniently, and perhaps with sufficient accuracy, 

 by means of a toothed wheel fixed on the axis of the frame, and running into a 

 pinion in the proportion of 10 to 1 : this pinion carrying a wheel which runs into a 

 second pinion in the same proportion, and so on through a train of wheels and 

 pinions, each wheel of which being divided into 10 equal parts, and numbered, will 

 show one digit of the decimal part of a revolution up to as many places of decimals 

 as there are wheels. The complete revolutions of the frame may in like manner be 

 registered by means of a pinion on the axis of the frame running into a wheel in the 

 proportion of 1 to 10. The value of N may thus be registered to four places of de- 

 cimals. 



Professor Moseley then proceeded to show — 1st, that if the edge of the rail had 

 the form of a straight line, or rather a curve, so slightly deviating from a straight 

 line as to give to the point of contact the same motion as that point would receive 

 from a straight line fixed upon the revolving plate, and actually passing through that 

 point, and if the perpendicular distance a of this line from the centre were such that 



:= "43429, and if the registration commence from that position of the plate in 



which the rail is perpendicular to the diameter traversed by its point of contact with 

 the wheel, then will the number registered be represented by the formula 



N=log,{tan(-l-+i-)} 



So that when a straight rail is thus fixed upon the plate, the machine may be made 

 to calculate the logarithmic tangent of any arc of the quadrant, and to replace a 

 table of logarithmic tangents. 



2nd. That if the edge of the rail were a circle whose circumference passed through 

 the centre of the plate, and whose radius was equal to t q, and if the angle 6 were 

 measured from that position in which the diameter of this circle coincided with the 

 line traversed by the point of contact, then would the number registered be repre- 

 sented by the formula ' i., -a 



N = sm 0. 



So that with this form of rail the machine would serve to calculate mechanically the 

 natural sines of angles, and to replace a table of such sines. 



3rd. That if the edge of the rail were a curve slightly differing from an ellipse 

 having its centre in c, and such that the point of contact of the plate and integrating 

 wheel might be made to move precisely as it would if guided by the actual pressure 



• See Poisson, Journal de I'Ecole Poly technique, ISme cahier, p. 320 ; or Art. 2. in the 

 Treatise on Definite Integrals, in the Encyclopsdia Metropolitana, by Prof. Moseley. 



