2 Mr. S. H. Burbury on the Law of Probability 



(j>(^y) is not expressible as the product of two functions, 

 one of x only, the other of y only. 



2. It follows from this definition, that if x and y are both 

 functions of any the same variables, x and y are correlated. 

 If for instance (1) f(x, t) — 0, and (2) <f)(y, £) = 0, so that x and 

 y are both functions of t, f\(x) of art. 1 is by (1) a function 

 of t as well as of x, and therefore by (2) a function of y as 

 well as of x ; and therefore x and y are, by the definition, 

 correlated. Generally, if n variables x ± . . .x n are connected by 

 equations of condition less than n in number, correlation 

 exists between them by virtue of those equations. It follows 

 also that if x and y are correlated, any functions of x and y, 

 as f(js) and <f>(y), are correlated with each other. 



3. Of the mean product of two correlated variables. — The 

 most general definition of mean product is this : There being 

 N values of x, and N values of y, we assign to every x some 

 one of the N values of y as a companion factor, by this means 

 forming N products each of an x and one y. Let them be 



*iyii ^2, • • • %?/ N - Then I define xy = X ^l — :___f^ # 



This selection of products might be effected in any one of 

 JN different ways. Practically, if x and y are both functions 

 of a third variable t, we might take every x with the value 

 of y for the same t, so that xy=§xy dt. Similarly, if x and y 

 are functions of two other variables z and t, we should define 

 xy ~ jj xydzdt, and so on. 



4. The square of the mean product, so defined, of two 

 correlated variables, cannot be greater than the product of 

 their mean squares. For, taking the general definition of 

 mean product above given, there are, as there stated, \N dif- 

 ferent ways in which N x's and N y's may be arranged to 

 form N products. Of these there must be some one way for 

 which xy is not less than for any other. And for this one, 

 and therefore for every other way, 



^' 2 - 2 _2*/ 2 ™_2^ 



and 



x 2 . y 2 -(xy) 2 =^{(x l y r -X2ij l f + (x 1 y z -x 7i y i y+ &c. 

 which is >0. 



