36 KEPORT 1841. 



serves as a guide along which that wheel maj* be made to traverse in the direction of 

 a diameter of the plate, retaining always its position in a plane perpendicular to it. 



One of the axes about which the frame turns is hollow, and through it a rod passes 

 to the centre of the integrating wheel, through which it passes, being so attached to 

 it (by means of a shoulder and nut) that the integrating wheel may turn freely on its 

 extremity, whilst any motion communicated to it in the direction of its length may 

 cause the wheel to move along the frame or its point of contact with the circular plate 

 to traverse one of its diameters. This rod is fixed firmly at its other extremity to a 

 short vertical piece, which connects it with another rod returning parallel to the first 

 between it and the plate, and passing through an aperture in the same piece (sliding 

 in vertical grooves) in which the hollow axis of the first-mentioned frame is fixed, 

 until it nearly reaches the integrating wheel ; its extremity is there turned down- 

 wards at a right angle until it just touches the surface of the plate. On the surface 

 of the plate is elevated a curved edge or rail of a certain geometrical form, determined 

 by the particular function which the machine is intended to integrate ; and against the 

 edge of this rail that small portion of the extremity of the last-mentioned rod which 

 is turned downwards, is kept continually pressed by means of a spring acting against 

 the vertical piece which forms the extremity of the frame composed of the two par- 

 allel bars, and which was last described. 



The circular plate has teeth cut upon its edge, and is made to revolve horizontally 

 by means of an endless screw fixed to the frame on which the plate rests, and thus 

 revolving it carries round the integrating wheel by reason of the friction produced 

 between its surface and the edge of that wheel by the weight which rests upon the 

 latter, and is supported at their common point of contact ; whilst the integrating 

 wheel is thus made to revolve, its point of contact with the plate is made (at the same 

 time) to traverse in the direction of a diameter to the plate, by the continued pres- 

 sure of the edge of the curved rail on the extremity of the lower rod of the last de- 

 scribed frame of two parallel rods ; the whole of that frame, and therefore the upper 

 of the two rods which compose it, being made by that pressure (the rail not being a 

 circle concentric with the plate) to move in the direction of its length, and the inte- 

 grating wheel to which the upper rod is attached being thereby made to slide along 

 the first-described frame, which frame at the same time it carries round by the wheel. 



Now it is evident that the point of contact of the integrating wheel with the plate 

 being thus made to alter its position on a diameter of the latter continually by the 

 pressure of the rail, as the plate to which it is affixed is made to revolve, the geo- 

 metrical law of this change of position must be dependent upon the polar equation 

 to the curve of the rail from the centre of the plate, or upon the relation of its radius 

 vector (the centre of the plate being taken as the pole) to the angle which that fine 

 makes with any line given in position and dmwn through the centre of the plate. 



Now let it be conceived that such a geometrical form is given to the curved edge 

 of the rail as to cause the distance of the point of contact of the wheel and plate from 

 the centre of the latter to be a given function /(^) of the angle 6 described by the 

 plate from any given position. The integrating wheel thus continually varying its dis- 

 tance from the centre of the plate, and its circumference continually revolving with the 

 motion of that part of the surface of the plate with which it is in contact, it follows 

 that the number of revolutions, and parts of a revolution, which are made by it, and 

 therefore by the frame which it carries with it as the plate evolves from any given an- 

 gular position 6\ to any other 6.2, is represented by the definite integral 





f{6)dd. 



For the distance of the point of contact of the integrating wheel and plate from the 

 centre of the latter being represented by f{6), it is evident that, whilst the exceed- 

 ingly small but finite increment A ^ of the angle 6 is described by the plate about its 

 centre, its point of contact with the wheel is made to describe an arc represented 

 by/(^) . A ^ ; so that if in any position of the integrating wheel its distance from 

 the centre of the plate be supposed to remain imchanged\y\i\\%t the small angle A ^ is 

 described by the plate, the circumference of that wheel will be made to describe a 

 space represented by the same quantity f{0).A&; and that the sum of all such spaces 



