189 
respectively the imaginary coefficients A and 7; or, more ex- 
actly, after the restoration of the powers of 7 and £ suppressed 
in virtue of the equations 7?=h?=—1; the imaginary coeffi- 
cients —j*k and — kj. Now, by the theorems in p. 170, the 
mean value of the product of two js and one & is, 
2 l 1 
ae peer =. 3 I)k=-5h 
and the mean value of the product of two 4s and one j is - 4). 
So that, so far as concerns the terms w*y*z* and a*y*z*, the 
difference between the two developments consists in this: that 
in (B) these terms are multiplied by ordinary products, but 
in (A) by mean products of js and ks. 
‘<The next discrepancy noticeable is in the coefficient of 
zy*z*, In (B) this is 7?h?, if the suppressed powers of 7 and 
k be restored. Now the mean value of the product of two js 
and two 4s, by the formule of p- 170, is 
ZW Ve 2a. 
je be gibt BS 
Here again we find a mean product of js and & s in (A), cor- 
responding to an ordinary product in (B). 
‘<The next discrepancy occurs in the case of the coeffi- 
cients of 2°y*z* and ay*z*. In (B) these are —7%h?, and 
— 7k, if we restore the suppressed powers of j and k. Now the 
mean value of the product of three js and two & s is, 
aCe OMarer eT 
and the mean value of the product of two js and three fs is 
1k. Here again, therefore, we find mean values of products 
of js and ksin (A), corresponding to ordinary products 
in (B). 
*« Let us next consider the coefficients of 2°y%z? and wy?z*, 
In (B) they are —j*k and — jk: but the mean values of pro- 
