ON amsler's planimeter. 405 



Now let it be assumed that the tracer T is moved from C to D ; the result 

 will bo that during the motion the block M will gradually pass along the 

 groove until the time when the tracer T has reached D ; and then, as the 

 length of the rod Q is exactly 5 inches, equal to the length of the side C D 

 (5 inches), the block M must have passed along the groove until the centre 

 N in that block is immediately over the point C, and the centre line of Q is 

 coincident with the line C D. If, now, the tracer T be moved along the 2 

 inches from D to A, the block M must move parallel with it, and the axis Q 

 of the wheels R, E,', R" will therefore be at right angles to the line of motion, 

 and the wheels themselves will, like the cylinder A in A A* of fig. 2, have a 

 rolling motion, and a rolling motion only ; and thus by the time the tracer T 

 has reached the point A, these wheels will each have made an entire revolu- 

 tion. If, now, the circumference of R or R', R" has been divided into ten 

 equal parts, and if on setting out from D pains had been taken to put the 

 wheel R with its zero mark to the pointer S, it would be found, on the arrival 

 of the wheel at A, that it had made an entire revolution, and that therefore 

 the index would read 10, equal 10 square inches— viz. the multiplication of 

 the length of the radius Q (5 inches) into the circumference of the wheel R 

 (2 inches). 



Now let it be assumed that the implement is to be used for the purpose of 

 measuring another rectangle A B C D, also of 10 inches area, having its sides 

 and ends respectively 2 inches and 5 inches long ; so that in this instance 

 (see fig. 4) the ends have the 5-inch measurement in lieu of the 2, and the 

 sides have the 2-inch in lieu of the 5. Once more let the tracer T be moved 

 from C to D ; the block M will now have only passed along the groove a 

 comparatively insignificant distance towards C, and the rod Q, will lie at the 

 angle shown, so that it will form the hypotenuse (5 inches long) of a 

 triangle of which the base wiU be C D (2 inches long). If, now, the tracer _T 

 be moved from D to A (5 inches), the block M will make a similar motion in 

 the groove ; and when the tracer T has reached A, the rod Q wiU have 

 moved parallel to itself, and will be found in the position shown in fig. 5. 

 But, as has already been said when speaking of A A^ of fig. 2,_if a cylinder 

 capable of rotating be caused to move over and in contact with a surface 

 when it is in a position neither parallel with, nor at right angles to, the line 

 of motion, and if it be made to preserve its own parallelism, the result will 

 be a motion compounded of sliding and of rolling, and the amount of the 

 rolling will bear such a relation to the whole motion as the base m n bears 

 to the hypotenuse n o. In the instance, therefore, under consideration the 

 ratio of revolution to the whole motion will be that of 2 to 5 ; therefore if the 

 zero on the wheel R were brought to the pointer S at the time of setting 

 out from D, it would be found, when the tracer had arrived at the end A of 

 its 5-inch journey D A, that the wheel R would have made just one revolu- 

 tion, and that the figure 10, indicating 10 square inches, would present itself. 



From a consideration of the foregoing two cases, it will be seen that the 

 " rate " of rotation of the wheel R, when it moves along the line D A, depends 

 upon the length of the line CD, and the "quantity" of such rotation upon 

 that of the line D A. These two expressions, " rate " and " quantity," will 

 be used hereafter in the above senses. 



As an illustration of " rate " and " quantity," suppose that the rectangle of 

 fig. 3 had only been half as long as the one that has been considered, namely 

 2^ inches, and had been bounded by the hue D' A' ; if, then, the tracer had 

 been moved from 1)' to A', the " rate" of revolution of the wheels R <isc. 

 would have been one half of the total distance moved through by the tracer, 



