THE METHOD OF LEAST SQUARES. 61 



The assumption expressed by 



is therefore either a simple mistake or a petitio principii: the 

 former, if it is deduced from the general principle that the pro- 

 bability of a compound event is equal to the product of those 

 of its elements ; the latter, if it is made to depend on the parti- 

 cular form assigned to /(a? 2 ). 



After all, too, if the demonstration were right instead of 

 wrong, it would not prove what is wanted. For if the law of 

 probability of a deviation parallel to a fixed axis is expressed 

 by the function 



-**&, 



VTT 



which is what the amended demonstration tends to show, the 

 probability that the stone falls on the area dxdy is plainly 





Transforming this to polar co-ordinates, and integrating from 

 to 2-7T for the angle vector, we get 2h 2 e~ h * r * rdr for the pro- 

 bability that the deviation from the mark lies between r and 

 r + dr ; a result which may be verified by integrating for r 

 from zero to infinity, the integral between these limits being 

 equal to unity. Thus if the deviations measured parallel to 

 fixed axes follow the law which the reviewer supposes to be 

 universally true, the deviations from the centre or origin fol- 

 low quite another; and hence it appears that his illustration 

 is altogether wrong. For if 2Ji 2 e~ h2r2 rdr is the probability of 

 an error lying between r and r + dr, the centre of gravity of 

 the shot-marks is not the most probable position of the wafer. 

 So that his hypothesis is self-contradictory. 



The original source of his error was probably the analogy 

 between Gauss's law, and the limiting function in Laplace's 

 investigation. 



I am, my dear Sir, 



Most truly yours, 



K. L. ELLIS. 



BRIGHTON, Sept* 19. 



