﻿112 Prof. 0. Henrici on a 



of a Fourier Series, 



y = £ A -f A x cos + A 2 cos 20 + . . . + A n cos n0 + ... 

 + B l sin0+B 2 sin 20 + .. . + B n sin n0 + . . . 



where 0= — -. 

 c 



The absolute term £A equals the mean ordinate of the 



curve, and can therefore be determined by any planimeter. 



It is the object of the Harmonic Analyser to determine the 



other coefficients. Their well-known values are 



1 C 2lT 1 C 2lt 



A n =- | y cos n0d0; B n = - I y sin n0d0. 



IT Jo 7T J0 



The Analyser is therefore an integrator. 



If the paper with the curve be wrapped round a cylinder, 

 the ordinate y falling on the generating lines or edges, the 

 axis of x along a circumference, then the curve will run back 

 in itself and form one continuous line, provided the circum- 

 ference of the cylinder equals the base c. That edge which 

 passes through the initial point A' of the original curve may 

 be called the zero-edge. 



Suppose the cylinder to lie horizontal with the zero-edge 

 at the top, then our angle will be the angle through which 

 the cylinder has to be turned in order to bring that point P 

 to the top which corresponds to any given x. Each edge 

 contains one point on the curve, excepting in case of a dis- 

 continuity where a finite length of the edge belongs to the 

 curve. 



§ 2. The first instrument of this kind was constructed by 

 Lord Kelvin (Proceedings Roy. Soc. vol. xxiv., 1876). Since 

 then several others have been devised. With regard to these 

 I may refer to my article " Ueber Instrumente zur harmon- 

 ischen Analyse " in the Catalogue prepared by Prof. W. Dyck 

 of Munich for the Mathematical Exhibition which was held 

 last summer in Munich, and also to the descriptions in the 

 Catalogue of the various instruments exhibited. 



These instruments differ essentially either in the manner in 

 which the trigonometrical factor is introduced, or in the 

 arrangement by which the actual integration is performed. 

 Lord Kelvin uses for the latter purpose his brother's disk- 

 globe and cylinder integrator, whilst a simple harmonic 

 motion introduces the trigonometrical factor. Sommerfeld 

 and Wiechert * of Konigsberg make the cylinder on which 



* See above Catalogue, p. 274. 



