191 
D> yzi=j21-kyz, 
Db? yz? =-2lyz, 
De yzt=— 2) 92343! hye, 
Dt yzt=4! y2°, 
De yzt=4! j2°-5! kyz, 
b§ yz'=-6! yz", 
Hence, we find 
3 
w yz =ye)+ a (jet — kyz*) - ayz? - a (2! jz%-3! hyz*) 
. A) 
5 ( 
+ avtyz + A (41 j25-5! kyz*) - ayz" — &e. 
Now let us compare this with the development of 
(y+ja) (z+kn). 
Expanding by the binomial theorem, and preserving the 
powers of j and 2, when both appear in the same coefficient, 
we have 
(y+ je) (2+ hey? = ye + a (ja? — hye) — a" (jhe + yz") 
+ 08 (jh?23 + hyz*) — at (jh824 — yz) 
+25 (phi — ky2*) — a8 (jh’e*+yz7) .. (®B) 
The discrepancies between the developments (A) and (B) are 
numerous, but all of them are of the same kind. In the first 
place the terms a°z*, atz+, a'z*, &c., do not appear in (A). 
In (B) they have the coefficients jh, jh, jh®, &c. But the mean 
values of such products of 7 and & are equal to zero. 
*«‘ Again, the mean value of the product of one j and 2» 
: 1 = ie 
ks is, a1 (— 1)”. Hence the coefficients in the two develop- 
Vv 
ments of a°z*, wz, &c., differ just in this: that in (A) they 
are mean products, in (B) ordinary products of js and ks. 
Thus it appears, as we anticipated, that if we substitute mean 
products of js and ’s for ordinary products throughout the 
