ON amsler's planimeter. 407 



assume that the height of the rectangle had been halved, and that it had been 

 bounded by the lines C D, D" B\ then the wheels 11 &c. in traversing from D 

 to D" would do so at their full " rate " of revolution, the line C D being 5 

 inches long ; but the " quantity " of such revolution would only be half that 

 AvMch it was in going from J) to A, because D D" is only half D A, and 

 therefore the wheels again would register but a half revolution, indicating 

 truly the 5-inch area of the 5-inch by 1-inch parallelogram D D", B' C. 



In each of the foregoing cases it has been assumed that the index is read 

 when the apparatus is about to start from D, and is re-read when it reaches A. 

 Such a reading would be quite sufficient in the case of a rectangle where the 

 groove is assumed to be in the prolongation of one of the sides (E C) ; 

 but under any other circumstances the complete circuit of the figure must be 

 made. To test this, let it be assumed that the tracer T starts from C, and that 

 the index on 11 is read just before the starting, and then let it be examined 

 when the tracer T has reached D ; it will bo found that the wheel R has 

 received an amount of rotation approximately that due to its ti'aversing the 

 arc of the radiiis N R, that R' has received a larger amount of traverse, and R" 

 a still larger amount, owing to their greater distance from the centre N ; but 

 it will be afterwards found that these amounts of revolution may be wholly 

 neglected, and that they will not come into the final computation, because, 

 assume the tracer T to have attained to the point A and to have put into the 

 wheels R, R', R" the one revolution which it has been seen that traverse would 

 give, those wheels would be found at A (were there any means by multi- 

 plying gear, as in the actual machine, to record more than the one revolution) 

 to have made the one revolution each, plus the varying amounts of revolution 

 which they would have received in their journey from C to D. But in their 

 back journey from A to B it is manifest they will each of them unwind (if 

 such a phrase may be used) exactly the quantity of revolution which was 

 put into them in moving from C to D. Further, during the passage from B 

 to C to complete the circuit, the direction of motion being parallel with the 

 position of the rod Q, the axle of the wheels R, R', R", no rolling movement 

 will be communicated to them, as they will be in the condition of the cylinder 

 A A^ of fig. 2, and will merely shde over the paper, so that on the arrival of 

 the tracer T at C, having made the circuit of the rectangle, there will be 

 found in them the one revolution, and neither more nor less than the one 

 revolution, generated by the traverse from D to A. 



The next point to be proved is the manner in which the implement wiU 

 truly record if the groove be not on the line produced by prolonging one 

 side of the rectangle. Let fig. 6 represent a rectangle, say 2 inches long on 

 its side C D and 5 inches high at its end D A, and containing therefore 10 

 square inches, and let X Y be a line parallel with B C, and as far removed 

 (2 inches) on the right hand from it as D A is removed from it on the left 

 hand, and let the groove be on the line X Y ; then, if the tracer T were 

 to stand at C, and the wheels R ifec. were at zero, and if the tracer were 

 then moved along the line C B, there would be put an amount of revolution 

 into R which would be compounded of the " rate " duo to the length Y C 

 and of the " quantity " belonging to the length C B, or 2 multiplied by 5 

 equal 10 inches, equal one revolution of R. But if now the tracer T be 

 brought back again along the line B C, the wheel R will unwind the revolu- 

 tion that was put into it, and on its return to C will be found at zero. 



Having thus premised that during the passage of the tracer T from B to C 

 the wheel R will have unwound or made a negative quantity expressive of 

 the rectangle B X Y C, let the measurement of A B C D be considered. As- 



