189 
respectively the imaginary coefficients & and j; or, more ex- 
actly, after the restoration of the powers of j and & suppressed 
in virtue of the equations 7?=h?=-1; the imaginary coefh- 
cients —7?k and —k?j7. Now, by the theorems in p. 170, the 
mean value of the product of two js and one & is, 
2!1 
3, 
2!1 1 
3 (2, Dray I)k=—gk; 
_ and the mean value of the product of two &s and one 7 is — 4). 
So that, so far as concerns the terms 2*°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 
z'y%z3, In (B) this is /*h, if the suppressed powers of 7 and 
: k be restored. Now the mean value of the product of two js 
and two &s, by the formule of p. 170, is 
PAN pi ak | 
Acta a Whee 
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°y4z* and a*°y*z*. In (B) these are —7*h?, and 
— 7?h, if we restore the suppressed powers of jandk. Now the 
mean value of the product of three j s and two fs is, 
See edly 
Bre ein Be” 
and the mean value of the product of two js and three /s is 
a 1k. Here again, therefore, we find mean values of products 
_ of js and ks in (A), corresponding to ordinary products 
m in (B). 
‘* Let us next consider the coefficients of 5y*z” and a*°y*z°, 
_ In (B) they are —j*k and — jk‘: but the mean values of pro- 
