18 KAEL PEARSON 



We have here at r = l a typical instance of the discontinuity. 



In Table I. columns (i) and (ii) the calculated ordinates of <f> 3 and (f> s are given, 

 the latter having been determined by the use of Legendre's Tables of the Elliptic 

 Integral F. In these cases, as in the later values of the ordinates of the dispersion 

 curves, 2V is taken as unity. The dispersion curves are plotted in Diagrams I. 

 and II. The Rayleigh solution is given in broken line ; it will be noticed how 

 very far it is from representing the facts at this early stage of the number of 

 flights. One of the most interesting features of the investigation is to mark the 

 gradual approximation of the discontinuous series of functions to the Gaussian normal 

 curve of errors as the value of n increases. 



The first test of the graphical method of dealing with the problem was to 

 start from the curve for n = 2 and construct the graph of cj> 3 . The result was 

 found to be extremely close to the elliptic integral solution obtained by analysis 

 and calculated from Legendre, and this gave us every confidence in the correctness 

 within reasonable limits of the graphical solution, where no such direct verification 

 was possible. After the ordinates of any graph had been found their differences 

 were plotted, and these difference curves submitted to most careful inspection. 

 Larger irregularities led to a reinvestigation of the points, smaller irregularities were 

 smoothed with the spline, and from the final smoothed difference curve the ordinates 

 were corrected. 



Another test was now possible. In every case 2ir <f> n (r 2 ) r dr ought to be unity. 



Each ordinate was now multiplied by its r and a quadrature formula used to find the 

 integral. The integral would usually differ very slightly from unity. Its reciprocal 

 was then used as a factor to each ordinate and the ordinates so modified were 

 the final corrected ordinates of the corresponding graph. The graphs were made 

 on a large scale, and the accompanying Table I., columns (iii) — (vi), gives the 

 ordinates of the dispersion curves from four to seven flights. 

 Additional tests were as follows : 



Since <£ n+1 (r 2 ) = -^ f \ (r 2 + 1 2 - 2lr cos 6) dd, 



ATI J 2ir 



1 



it follows that <f> n+1 (0) = — fa (l>) dd = <f> n (1% 



2tt 



2tt 



or : The axial ordinate of the n+ 1th dispersion curve is the ordinate at a distance I, 

 or a flight, from the centre of the nth dispersion curve. Table IV. illustrates 

 the degree of accuracy reached here. 



The ordinate at r = I of the seventh curve given by the expansion in <u-functions is 

 '0375, and this is precisely the value of the central ordinate of the eighth curve given 

 by the same expansion. Thus the graphical method runs with surprising accuracy 

 into the analytical. The Rayleigh solution gives "0398 for the central ordinate of 

 the eighth curve as against the "0375 of the w-expansion, or the '0378 of the 



