628 PROFESSOR BOOLE ON THE COMBINATION 



mined being, not that of the arithmetical mean, but rather a principle of geome- 

 trical consistency, intimately connected with our ideas of the composition of 

 motion. 



The principle was first stated in a popular and somewhat inexact form, by Sir 

 John Herschel, I believe, in the Edinburgh Review.* It was afterwards made 

 the subject of an adverse criticism in the Philosophical Magazine, by Mr Leslie 

 Ellis. f There is no living mathematician for whose intellectual character I 

 entertain a more sincere respect than I do for that of Mr Ellis ; and even while 

 stating the grounds upon which I differ from him, with respect to the value of 

 Sir John Herschel's principle, I avail myself of his labours, in giving to that 

 principle a more scientific form and expression, and in developing its consequences. 

 The language adopted in the following statement, will be, as far as possible, that 

 of the author of the principle, — the analysis will be that of Mr Ellis. 



Suppose a ball dropped from a given height, with the intention that it shall 

 fall on a given mark. Now, taking the mark as the origin of two rectangular 

 axes, let it be assumed, that the actual deviation observed is a compound event, 

 of which the two components are the corresponding deviations measured along 

 the rectangular axes. Grant, also, that the latter deviations are independent 

 events. Further, let us represent by / (or), f (y 2 ), the probabilities of the respective 

 component deviations measured along the axes x and y, — we give to them this 

 form, because, positive and negative deviations being equally probable, the func- 

 tion expressing probability must be an even one, i.e., must not change sign with 

 the error. Hence the probability of the actual deviations observed will hef(x 2 ) 

 /(j/ 2 ). Let it be observed that this is not the probability of a deviation to the 

 extent \/x 2 + y 2 from the mark, but of a deviation to that extent in a particular 

 line of direction. Now, let the principle be assumed, that this expression is inde- 

 pendent of the position of the axes, i.e., that we may regard component deviations 

 along any two rectangular axes as independent events, by the composition of 

 which the actual deviation is produced. We have then x and y representing 

 two new component deviations, 



/(^)/(2/ 2 )=/M/(2/' 2 ) ... (2) 



If y'=\/x 2 +y 2 then ^=0 and we have 



/(^)/(2/ 2 )=/(0)/(^ + 2/2) .... (3 ) 



An equation of which the complete solution is, 



A and h being constants. The condition that the probability of the error must 



* Vol. xcii. p. 17, Art. Quetelet on Probabilities. 



f Vol. xxxvii. p. 321, " Letter addressed to J. D. Forbes, Esq., Professor of Natural Philosophy 

 in the University of Edinburgh, on an alleged proof of the Method of Least Squares." 



