ON QUADRATURES AND INTERPOLATION. 377 



In the usual form of the instrument, the reciprocating curve, traced 

 by the other end of the bar, is an arc of a circle. This is for facility of 

 use and construction, and is by no means essential. 



Amsler's Meclianical Integrator. — By an ingenious extension of the 

 principle of his planimeter. Professor J. Amsler-Laffon, of SchafEhausen, 

 has consti'ucted a machine which, while a pointer describes a closed curve, 

 records simultaneously its area, its statical moment, and its moment of 

 inertia about a given axis. In this case the butt end of the bar which 

 carries the pointer is made to move along a right line. The area is read 

 off fi'om a wheel mounted on the bar itself, and this part of the operation 

 is thus the same as in the common planimeter. The moments are read off 

 from wheels mounted on arms whose centres also describe right lines, but 

 which are so geared with a wheel rigidly connected with the bar carrying 

 the pointer, as to turn relatively to it with the fixed velocity ratios of 

 2 : 1 and 3:1. Supjiosing the angular motion of the pointer-bar to be 

 y, and the velocity ratio n : 1, the quantity of rotation of the second 

 circle will he nQ + a, a being an arbitrary constant depending upon the 

 initial position. When the pointer goes round any closed curve which 

 does not contain the centre of the first circle, this measurement of this 

 rotary motion comes to nothing, for the angular movement is the same 

 forward as backward, and it may therefoi'e be left out of account. But 

 its angle {n Q + a) settles the direction of the resolution of which the 

 component is measured by the instrument, when there is linear motion of 

 the centre. The rolling wheel records a constant multiple of 



— dx cos (?i Q + a), 



where dx represents the movement parallel to the axis, and its complete 

 record is 



— / dx cos ()i 4- o) = — 



u 



taken over the whole area of the curve. This has to be multiplied by a 

 numerical factor, which is one of the constants of the instrument. 

 Where »i := 2, if we make a = 0, we have 



ii= r dx cos 2 = /^ dx 1 1 - 2 (sin Oy \ = C dx (1-2 1^ ) 



y being an ordinate perpendicular to the axis of x. 



In this case therefore the difference between two readings of the 



rolling wheel counter is always proportionate to / t/^ f?.i', which thus gives 



the statical moment. 



When n = 3, if we make a = — g ^r, 



we have 



cos (S - ^ff ") = sin 3 = 3 (sin 0)3 - 4 sin 



and therefore the reading given by the wheel is 



/ ay^ dx — I hy dx 



