1878.] Sir W. Thomson. Harmonic Analyzer. 



371 



III. A curve of order n> in flat space of 11 dimensions (and no less), 

 is always unicursal. 



From this the author obtains a representation of the points of an 

 n dimensional space by means of groups of n points on such a unicursal 

 curve, corresponding to the methods of Hirst and Darboux for 

 3-dimensional space. 



When n is even, the system corresponds to that of poles and polar s 

 in regard to a quadric locus upon which the curve lies. 



When n is odd, every point is co-flat (i.e., n + 1 points lie in the 

 same n — 1 flat), with the n points of the osculant (n — 1) flats which 

 can be drawn through it. 



IV. Every curve of order n in flat space of n— 1 dimensions is either 

 unicursal or elliptic. 



V. When the curve is unicursal, and n is odd, the n points of super- 

 osculation, or points of stationary osculant (n — 2) flats, are on the 

 same (n — 2) flat. But when n is even, this will be the case only 

 under a certain condition. 



VI. When the curve is elliptic (or bi-cursal) the cla'ss of the curve 

 is n (n — 1), and the number of superosculants n*. 



If we consider a curve of the order n and deficiency p, existing in h 

 dimensions, a (k — 1) flat cuts such a curve in n points, such that the 

 sum of each of the p parameters (Abel's theorem gives p equations be- 

 tween the parameters of w points which lie on a (Jc — 1) flat), for these 

 n points is zero. And we obtain the theorem 



VII. A curve of order n and deficiency p, not greater than \ can 

 at most exist inn — p dimensions. 



IV. "Harmonic Analyzer." Shown and explained by Sir 

 William Thomson, F.R.S., Professor of Natural Philosophy 

 in the University of Glasgow. Received May 9, 1878. 



This is a realization of an instrument designed rudimentarily in 

 the author's communication to the Royal Society (" Proceedings," 

 February 3rd, 1876), entitled "On an Instrument for Calculating 

 (f<p(x)-f(x)dx), the Integral of the Product of two given Functions." 



It consists of five disk globe and cylinder integrators of the kind 

 described in Professor James Thomson's paper " On an Integrating 

 Machine having a new Kinematic Principle," of the same date, and 

 represented in the annexed woodcuts. 



The five disks, } are all in one plane, and their centres in one line. 

 The axes of the cylinders are all in a line parallel to it. The diameters 

 of the five cylinders are all equal, so are those of the globes ; hence 

 the centres of the globes are in a line parallel to the line of the 

 centres of the disks, and to the line of the axes of_the cylinders. 



