264 Prof. J. Thomson on [Feb. 3, 



on the cylindric surface and the other on the disk or cone, when both 

 considered as lines traced out in space fixed relatively to the framing of 

 the whole instrument, will be two parallel straight lines, and that the 

 line of motion of the ball's centre will be straight and parallel to them. 

 For facilitating explanations, the motion of the centre of the ball along 

 its path parallel to the axis of the cylinder may be called the ball's longi- 

 tudinal motion. 



Now for the integration of ydx : the distance of the point of contact 

 of the ball with the disk or cone from the centre of the disk or vertex 

 of the cone in the ball's longitudinal motion is to represent y, while 

 the angular space turned by the disk or cone from any initial position 

 represents oc ; and then the angular space turned by the cylinder will, 

 when multiplied by a suitable constant numerical coefficient, express the 

 integral in terms of any required unit for its evaluation. 



The longitudinal motion may be imparted to the ball by having the 

 framing of the whole instrument so placed that the lines of longitudinal 

 motion of the two points of contact and of the ball's centre, which are 

 three straight lines mutually parallel, shall be inclined to the horizontal 

 sufficiently to make the ball tend decidedly to descend along the line of 

 its longitudinal motion, and then regulating its motion by an abutting 

 controller, which may have at its point of contact, where it presses on the 

 ball, a plane face perpendicular to the line of the ball's motion. Other- 

 wise the longitudinal motion may, for some cases, preferably be imparted 

 to the ball by having the direction of that motion horizontal, and having 

 two controlling flat faces acting in close contact without tightness at 

 opposite extremities of the ball's diameter, which at any moment is in 

 the line of the ball's motion or is parallel to the axis of the cylinder. 



It is worthy of notice that, in the case of the disk, ball, and cylinder 

 integrator, no theoretical nor important practical fault in the action of 

 the instrument would be involved in any deficiency of perfect exactitude 

 in the practical accomplishment of the desired condition that the line of 

 motion of the ball's point of contact with the disk should pass through 

 the centre of the disk. The reason of this will be obvious enough on a 

 little consideration. 



The plane of the disk may suitably be placed inclined to the horizontal 

 at some such angle as 45° ; and the accompanying sketch, together with 

 the model, which will be submitted to the Society by my brother, will 

 aid towards the clear understanding of the explanations which have been 

 given. 



My brother has pointed out to me that an additional operation, impor- 

 tant for some purposes, may be effected by arranging that the machine 

 shall give a continuous record of .the growth of the integral by intro- 

 ducing additional mechanisms suitable for continually describing a curve 

 such that for each point of it the abscissa shall represent the value of 

 .r, and the ordinate shall represent the integral attained from x=0 



