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 
Joun Herscuet, I believe, in the Edinburgh Review.* It was afterwards made 
the subject of an adverse criticism in the Philosophical Magazine, by Mr Leste 
Euuts.¢ There is no living mathematician for whose intellectual character I 
entertain a more sincere respect than I do for that of Mr Exuis ; and even while 
stating the grounds upon which I differ from him, with respect to the value of 
Sir Jonn Herscuew’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 Extis. 
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 f(2’), f(y’), the probabilities of the respective 
component deviations measured along the axes 2 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, 7.¢., must not change sign with 
the error. Hence the probability of the actual deviations observed will be f(a’) 
f(y’). Let it be observed that this is not the probability of a deviation to the 
extent /x?+y? 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, 7.¢., 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 w and 7’ representing 
two new component deviations, 
(Cpt ACD Ce tA en (> 
If v=V7 2? +y? then a’=0 and we have 
S@)FSY=fO F(t). . . .  &) 
An equation of which the complete solution is, 
F@Q=Ke 
A and / being constants. The condition that the probability of the error must 
* Vol. xcii. p. 17, Art. QurTeteT on Probabilities, 
t Vol. xxxvii. p. 321, “‘ Letter addressed to J. D. Forzes, Esq., Professor of Natural Philosophy 
in the University of Edinburgh, on an alleged proof of the Method of Least Squares.” 
