128 TRANSACTIONS LIVERPOOL BIOLOGICAL SOCIETY. 



points, and in that case it will be seen that there are now 

 two places where the summational curve is concave, and other 

 two where it is convex to the horizontal axis. The frequency 

 series from which the summation has been made must now be 

 regarded as consisting of two simple, superposed series. 



In doing all this we are adopting a definite formula of 

 interpolation. In making the smoothed curve pass evenly 

 among all the plotted points we are making its total area equal 

 to the total area of the polygon formed by connecting the 

 plotted points by short, straight lines : in the Pearson method 

 of calculating a theoretical curve we must first assume that 

 the total unsmoothed frequency shall be equal to the total 

 smoothed frequency. Further, the equation of the curve which 

 is thus calculated by Pearson's methods is graphically repre- 

 sented by a certain form, and this form, for our smoothed, 

 summational curve, is given by the line of gradually changing 

 curvature, which is everywhere as near as possible to the plotted 

 points. The elasticity of the spring (which we assume to be 

 the same everywhere in it — not always the case, however, in a 

 much-used one) confers on the curve this gradual, unforced 

 change of curvature. 



The actual, smoothed curve, which is to be used further, is 

 drawn with a ruling-pen filled with red ink, and obviously a 

 fine trace is made. The points where the curve intersects the 

 vertical scale lines of the graph paper are now pricked and the 

 ordinates are read off on the vertical scale. These are written 

 down to replace the unsmoothed Sf series. This smoothed 

 series is next differentiated so as exactly to reverse the process 

 of summation, as it was carried out on the unsmoothed/ series, 

 and the result is a smoothed, frequency curve. It is not quite 

 smooth, however, because it is rather difficult to read off the 

 ordinates very precisely on the vertical axis. There is never 

 any difficulty, for all that, in drawing a smooth, frequency 

 curve through the points found by this process, for we can 



