24 KARL PEARSON 



Dispersal Curve for Five Flights (Diagram IV.). 



All infinite densities have now finally disappeared. The density vanishes at 

 the end of the range, but the dispersal curve makes a finite angle with the horizontal 

 axis. There is a marked discontinuity of slope at r = I ; a still more noteworthy 

 feature is that from r = to r = l the graphical construction, however carefully 

 reinvestigated, did not permit of our considering the curve to be anything but 

 a straight line. If this could be verified from the analytical expression 



N f°° 

 fa (r 2 ) =2^1 uJ o (ur) {J" (ul)}' du 



by showing that the integral is independent of r from to I it would be of much 

 interest. Even if it be not absolutely true, it exemplifies the extraordinary power 

 of such integrals of J products to give extremely close approximations to such simple 

 forms as horizontal lines. 



The approach of the Rayleigh curve to the result is now more noticeable. 



Dispersal Curve for Six Flights (Diagram V.). 



There is contact now of the first order at the end of the range. From r = 

 to r = l the curve of dispersal appears to be a sloping straight line tangential to 

 the continuous curve from r = I to r = 61. No other discontinuity of a low order 

 is now visible. The curve, except for the finite slope at r = 0, is becoming much 

 more of the Gaussian form. It runs fairly closely to the solution in oi-functions up 

 to cojss, in fact is not separable at the extreme part of the range, where the Rayleigh 

 curve still gives finite ordinates beyond the possible range. 



Dispersal Curve for Seven Flights (Diagram VI.). 



All sign of discontinuity has gone, the curve is horizontal at the centre of 

 dispersion and might be easily mistaken for a normal curve of errors. The expansion 

 in co-functions represents the result within the limits almost of constructional error. 

 It was not thought necessary to continue the graphical work beyond this stage. 

 We may conclude that : 



The deviation of the Rayleigh solution for seven and more flights from the 

 true dispersal curve is practically the same as its deviation from the solution in 

 co-functions when five terms of that series are retained. 



This I think completes the full solution of the fundamental problem. The dispersal 

 curves for the cases of 2 to 7 flights are given in the Table I. of ordinates and the 

 Diagrams I. to VI. For higher values the co-function series gives the solution. This 

 solution could be applied to calculate the ordinates of the dispersal curve for fewer 

 flights than 6 or 7, but several more w-functions would have to be used and the 

 arithmetical work — especially while these functions are as yet untabled* — then 

 becomes somewhat severe. 



* Table II. provides a preliminary series of values of the u-functions. 



