160 



TRANSACTIONS LIVERPOOL BIOLOGICAL SOCIETY. 



We have now two equations and by solving these 

 simultaneously we find the constants n and a. 



It may be worth while to give an actual example of the 

 calculation. 



1 



Days. 



X 



2 



Nos. of 

 bacteria 

 in unit 

 volume. 



y 



3 



Logs of 



X 



4 



Logs of 



y 



5 



Values 



of mid- 



ordinates 



6 



Bases 



of 

 trape- 

 zoids. 



7 



Areas. 



(5 x 6) 



a 



8 



Distance 



from 



arbitrary 



origin. 



d 



9 



1st 

 moments 



about 

 arbitrary 



origin. 



a x d 



y 



1 

 2 

 3 

 4 

 5 



2,310 



886 



43 



36 



2 



0-0000 

 0-3010 

 0-4771 

 0-6021 

 0-6990 



3-3636 

 2-9474 

 1-6335 

 1-5563 

 0-3010 



3-1555 

 2-2904 

 1-5949 

 0-9286 



0-3010 

 0-1761 

 0-1250 

 0-0969 



0-9498 

 0-4033 

 01994 

 0-0899 



- 0-1505 



- 0-1429 



3-58 

 2-47 

 1-82 

 1-35 



+ 0-088 

 + 0-2386 

 + 0-3495 



+ 00355 

 + 0-0476 

 + 00314 



1-6424 



+ 0-1145 

 - 0-1429 







- 0-0284 





Area of whole distribution 

 - 0-0284 



v = 



i 



1-6424. 

 - 0-0172 



1-6424 



Mean = arbitrary origin - 0-0172 = 0-3010 - 0-0172 = 0-284 

 Limits are therefore - 0-284 and 0-6990 - 0-284 = + 0-415 



Area = 1-6424 = 



r 



J x — 



(nx + a)dx = 0-0458 n + 0-699 a 



fix 







/ 



0-415 



X = 



[x(nx + a)] dx = 0-0314 n + 0-0458 a 



These equations give y = — 3-7# + 2-535 



We have now found the equation to the straight line 

 which fits best the irregularly placed points which we have 

 quoted. The area beneath this straight line, and between the 

 first and last ordinates, is equal to that formed by the series 

 of trapezoids. We calculate the new ordinates, — n being 



