190 
ducts of one j and four £s, or four js and one &, are respec- 
tively 
ifs! 
78 nd = aS. or gj and 5 k, 
so that here likewise we find mean products in (A) standing 
in place of ordinary products in (B). 
‘‘ Lastly, the coefficients of x°y*z*, and x'y*z* in (B) 
are j*k? and 7*k*. Now the mean value of the produgt of four 
js and two ks, or of two js and four /s, is = — 3, which is the 
coefficient belonging to a’y%z? and a°y*z* in (A). 
‘¢ It is, moreover, to be observed that all the terms which 
disappear out of (B) have coefficients like jk*, the exponents of 
both j and & being odd numbers. Now the mean value of a 
product containing odd numbers both of js and /s has been 
proved equal to 0. 
‘It is also deserving of remark, that where the developments 
coincide, the mean values and the ordinary products are equal. 
Th fact, these coincidences occur in the case of the first and last 
terms in each group of terms multiplied by the same power of 
x; and in their coeflicients js and /s are not combined. 
‘¢ So far, then, as our examination has extended, the discre- 
pancy between the developments (A) and (B) consists in this, 
that mean values of products of js and /s stand in the former 
in place of ordinary products occurring in the latter. 
‘«‘ The careful examination of this one example led me to 
suspect, that when m and n are integers, the difference between 
the expression 7 y™z”" and the ordinary algebraic develop- 
ment of (y + ju)™ (z+hka)”, effected without any regard to the 
properties of 7 and &, consists merely in this, that mean pro- 
ducts of js and ks take the place in the former of ordinary 
products occurring in the latter. To test this hypothesis let 
us try another simple example, in which y and z are not 
symmetrically involved. Let us calculate 7 yz1. We shall 
have then 
