158 TRANSACTIONS LIVERPOOL BIOLOGICAL SOCIETY. 



graphs go as best suits our argument. We ought to have a 

 method of making them which removes personal bias. 



An easy application of Pearson's method of moments 

 enables us to do this in the present case. 



Assume, then, that the general equation representing the 

 drop in numbers of the bacteria in the cultures is : — 



yx n = a 

 that is, the equation to a hyperbola. 



Now taking logarithms and transposing we get from this 

 equation : — 



log y = — n log x -f- log a, 

 that is, the equation to a straight line, — n being the tangent 

 which the line makes with the axis of x, and log a the intercept 

 on the axis of y ; we have now to find these constants n and a. 



If we plot these points given by the equation we find 

 that they are irregular, and we have the same difficulty in 

 drawing a straight fine through them as we had in the original 

 data. If the points lay very near a straight line we could 

 find both n and a by inspection of the graph. But since this 

 cannot be done with certainty, we have to find the constants 

 analytically. Let us regard the logs of y and x, not as logs, 

 but simply as y and x co-ordinates. They form a distribution, 

 and we have to find the mean of this, and the first moment of 

 all the frequencies in it about the mean. The graph of the 

 distributions shows us a series of trapezoids on unequal bases. 

 To find the moments we must therefore calculate the areas of 

 these trapezoids, and then the mid-ordinates. We must 

 suppose the areas to be concentrated round the mid-ordinates. 

 If the y- co-ordinates are y , y 2 , y 3 , &c, and the x- co-ordinates, 

 Xi, x 2 , x s , &c, we find the areas from : — 



while the corresponding mid-ordinates are given by 9 — - 1 



