ERRORS OF OBSERVATION. 413 



Secondly, let all the letters be of balanced values. Every collection of terms of one 

 type out of 2(a; + ...)'* now absolutely vanishes, if any one or more exponents be odd: and 

 every collection out of 2(a? + .,.)* now vanishes, if k be odd. The term of most letters in 

 2(»+..,)^* is that in which each exponent is 2, of all types that do not aggregately 

 vanish. The number of letters is Jc, and the multinomial coefficient is 1.2..,2A;:(1.2)* or 

 I.3...2A;- 1 X 1.2...fe Hence the average 2A;* power of the sum of values, which I shall 

 denote by A^^, is 1.3...2& - 1 .1.2...A; multiplied by the sum of all terms of the form 

 2:0!*. 2:y*... with k letters in each product. But 1.2.3...A; multiplied by this product is all 

 that is to be retained of (2:a?2 + 2:y' + ...)* or {l'.{a! + y + ...f^ or Al. Hence the 

 following theorem : — If all the values of each letter be balanced, and the number of letters 

 which take value be infinite, then ^Ij^ being the average 2A;"' power of all the values, we have 



^24= 1.3.5 2A- l.Al. 



As a verification, let each of the a quantities take the values + 1 and - 1. The sum of the 

 2i* powers of the values is o-'* + 1^(0- _ 2)''* + 2^(o- - 4/* + ... + o-^(- ct)^*, which is the 

 operation {E + E-'^)" performed upon 0-*. This is {2 + A* (l + A)->}''0^*, and its highest 

 term is that which has A**(l + A)"*, of the terms of which only A^*0** has value. The term 

 to be retained is therefore ^^2''~*A*0*, or, <r being infinite, a*2°'"*2. 3...2A; : 2.3...)fc, 

 or 1 .3.5...2^ - 1 . S'ff*. Dividing by 2% the number of values, and remembering that 

 the average square for each letter is 1, we see the verification of the theorem. 



We may now adjust our supposition to the problems of our subject by supposing that 

 each letter has an infinitely great number of continuous values, those infinitely near to v 

 entering proportionally to the element of an integral, cpvdv, so that the average k^^ power 



of values of this letter is j(pv.v''dv, taken from one extreme, as —J?, to the other, + E, 



If the extreme of integration give j<pvdv = l, then, all the or igip,al values being equally likely, 

 (pvdv represents the probability of a value taken at hazard lying between v and v + dv. 



Let (pv be called the modulus of facility of the value in question : I shall assume that 

 {(bv^f'dv, taken between extremes, is a finite quantity for every positive and integer value 



of k. The usual limits are — co and + co : I shall denote / by f. Should finite limits 



•'-to 



ever be in question, we must deduce out of our forms the consequences of supposing <bv a 

 discontinuous function of the form (- co ) (- i?) ^v (+ JB) (+ 00 ), where ± E are the 

 limits of error. 



Since (px is an even function, f (px . a;*+'da! = 0. And from this, and j ^x . a?^*d<r being 



always finite, it readily follows that ^'"'^a? . a?" vanishes when a; = ± OD , for all values of m and 



«, included. Remembering I (pxdx = 1, the following results are easily obtained, m, a, k, 

 being positive integers. 



