﻿114 Prof. 0. Henrici on a 



the planimeter will register a number proportional to the 

 coefficient A n or B n . 



I had an instrument of this kind made early in 1889, but 

 it did not turn out quite as simple as its theory. It gives, of 

 course, only one coefficient at a time, though it would not be 

 difficult to construct it to give more terms if it were not for 

 the mechanism required to produce the simple harmonic 

 motion. This always introduces a certain amount of friction 

 if it is to work accurately. I therefore tried to do away with 

 this, and obtained my object in the manner now to be 

 described. 



§ 4. If the definite integrals which determine the coefficients 

 A n and B n be integrated by parts, we get for the former 



2n r*2ir 



nir A n = [y sin n0^ — I sin n6 dy, 



o Jo 



the limits relating to 0. 



If the original curve is continuous, the integrated part 

 vanishes. This is not the case if there is a discontinuity, at 

 least not if is retained as the independent variable. 



To prove that in this case also the integrated part can be 

 neglected, let us consider the curve in fig. 2. Let & be the 

 value of for which the discontinuity C C occurs, and let 

 y{ be the ordinate of C, and y 2 ' that of (7. 



The integral with regard to has to be broken up into 

 two, the first going from to 0', the second from & to 2ir. 

 The integrated part, therefore, gives 



yi sin nO' —y 2 ' sin nd\ 



and this, in general, does not vanish. 



The remaining integral has to be taken for the two parts 

 of the curve from A' to C and from 0' to B', if the curve is 

 not made continuous. But if the curve is made continuous, 

 we have also to take the integral for the intervals from C to C, 

 and from B' to B". For these dd vanishes, but not dy. This 

 gives in addition the integrals 



f c ' . P c ' 



— I sin n0 dy= —sin n0' \ dy = —sin n& (yj—yi) ; 



hence just the terms obtained before from the integrated 

 part. 



The second integral for the interval B ; B" vanishes because 

 it is multiplied by sin 2n7r. In case of the coefficient B n this 

 is not the case, but then the integrated part also contains 

 more terms which equal it. Hence : — 



