412 Me DE morgan, ON THE THEORY OF 



a> /3> 7j whatever be its letters or suffixes. Hence, P being the coefficient of expansion of 

 ^aVb^^' ^^ ^"^ ^^^^ 'S^'^i/^z^, — where S refers to all terms of the type, from all the values 

 of (a? +...)* — has the coefficient PNiX/xv. Divide both sides by A'', and we see that the 

 multinomial theorem holds of averages. That is to say, if we expand {as + y + x +...)*, and 

 for each power of a>, y, &c. write its average, we have the average value of (x + y + z ...)*. 

 This theorem lies hid in many cases of multiple integration. 



Any number of values of a letter may be equal, so that by different sets of equal values, 

 forming parts of the whole set, any probabilities of occurrence of any amount of value may 

 be represented. If any letter have balanced values, that is, if — a occur as often as + a, 

 a being any one of the values, it is obvious that all the average odd powers of that letter 

 vanish, and all the average products into which any odd power enters. 



Thus if all the letters, or all but one, be balanced, we see that the average square of the 

 sum is the sum of the average squares. 



Let the quantities x, y, %,... increase without limit in number; but to avoid the prolixity 

 of the language of limits, let us say that the number is infinite ; and let the number be a. 

 As to the several values of the letters, tliose of any one may be finite or infinite in number ; 

 the results will be in no way affected by the transition from one supposition to the other. 



First, let all the values of all the letters be positive. A term having h letters, with 

 assigned exponents — of which the sum must be k — appears in each value of {x + ...)* in as 

 many ways as there are some sort of mutations of k out of o- : and this number is of the order 

 a . Accordingly, a being infinite, we need only retain the terms in which h is greatest ; and 

 the same after substitution of averages. Now h is greatest, and is = k, when each letter 

 enters only in the first power: and the multinomial coefficient is then \.2.S...k. Hence the 

 average A;*'' power of (.r + ...) is 1.2...& times the sum of all the products of the form 

 2:aj2:y 2:«... with k letters in each product. But, by the same reasoning, this is all that 

 need be retained of (2:a? + 2:y + ...)*. Hence the following theorem: — All values being 

 positive, and the number of letters which take value being infinite, the average of the A;"* 

 powers of the sum of values is the A"" power of the sum of average values. 



By way of verification, let each of the a letters be either or 1 : and let n^ represent the 

 number of combinations of n out of a. The sum of the A"* powers of all values of <» + y + ... 

 is 0* + 1, 1* + 2^2* + ... + (T„(j', which is the operation (l + Ef performed upon 0* ; where 

 EnJ' = (rre + l)*. This is the operation (2 + A)'' ; and A"0* vanishing when n> k, the highest 

 term is ^^2''-*A*0*, which, — since <r is infinite, and A*0* = 1.2...^, — is o-*2''-*. Divide by 

 2", the number of values, and the average /;* power of the sum of values is (iff)*, or 

 (^ + ^ +... 0- terms)*, or the A"* power of the sum of the averages. 



Another simple application of this theorem, easily verified by the integral calculus, is as 



/.» 

 follows. If (pa.dx, (b{a + dw).dx, he, the elements of / (pxdof, be multiplied together 



*'« 



k and k, the sum of all the products is i j cpaidx] : 1. 2... k. 



