376 EEPOET — 1880. 



follows the curve on the other side of the templet, the travel of th© 

 cylinder will effect the quadrature of the curve on the templet.* 



The use of the machine is by no means limited to this simple quad- 

 rature. By putting the fork in gear with the disk through the intervention 

 of suitable wheel or link- work, or belting; or by gearing two such 

 machines suitably together, it is possible to obtain the mechanical solution 

 of differential equations. f 



Sliding motion is not altogether escaped in this machine. In the first 

 place, there is sliding motion of the sphere in the fork. Further, the 

 rolling of the sphere along a small circle is not pure rolling, but, although 

 not having any actual sliding, is intermediate between sliding and rolling. 

 For if we separate the pure rolling along a great circle from the twist 

 necessary to make it describe a small circle, the aggregate of this twisting 

 is the same as we should get by turning the sphere through a definite 

 angle about an axis perpendicular to the disk ; but instead of being finite 

 slidmg, as it would be on this last supposition, it is in fact distributed 

 over a line instead of concentrated at a point. It is thus infinitesimal 

 at every point of the line, along which, however, there ceases to be pure 

 rolling. J 



There is also a source of error in the necessity of giving some clearance 

 to the fork, which would otherwise not slide on the sphere. This clear- 

 ance introduces a slight error every time the fork reverses its motion. It 

 IS, however, a constant error ; but it is always in the same direction, and 

 is not compensated on a double reciprocation. This is the chief drawback 

 to the machine, which is nevertheless a most valuable instrument. 



^??isZer'sjjZa)ime<er.— In this wonderful little instrument a pointer is 

 made to ran round the closed curve, which has to be measured, and a 

 little wheel, which partly rolls and partly slides, gives the area by th& 

 mere reading of its rolling motion. The main principle on which it de- 

 pends IS this : that if a finite right line moves in its own plane, the whole 

 area swept out by it is measured by the product of the length of the line,, 

 and by the sum of the components of the motion of the middle point 

 (resolved at every instant) at right angles to the line. A wheel turning 

 on an axis parallel to the line, and free either to roll or to slide on the 

 paper or plane, will effect this instantaneous resolution, and its reading 

 will integrate the required component. When one end of the bar makes 

 a complete circuit of a closed curve, coming back to the point from 

 which it started, while the other end reciprocates along an arc of any 

 fixed curve, Avholly external to the closed curve, the area of the latter is 

 given by the difference between the initial and final readings of the wheel. 

 In this case, moreover, the principle of the separation of the motions of 

 rotation and translation shows that the total reading of the friction wheel 

 ^yill be the same, if it be moved from the middle to any other point of the 

 line, or even of the line produced. The only adjustment required, there- 

 fore, is that the axis of the rolling wheel should be parallel to the bar 

 which carries the pointer. This freedom from adjustment is one of the 

 most valuable properties of the instrument. As a practical matter, the 

 accuracy of the results which it gives is quite equal to that of the very best 

 drawings which can be made. 



* See Boy. Soc. Frocccdingg, vol. xxiv. jd. 262, ' On an Integrating Macliine liaving 

 a new Kinematic Principle,' by Professor James Thomson. 



t See two_ papers by Sir Wm. Thomson at pp. 266 and 269 of the same volume. 



X The motion is intermediate, in much the same sense that a-" (log x'f is inter- 

 mediate in dimension to x" and x" + ^. See De Morgan's Biff, and Int. Calc. p. 323. 



