183 
el 
Al ply! 
as, I have no doubt, you had determined it to be. 
{A, Ia v}= 
«« With the same signification of { }, we have, by (2), ° 
Ni, m, n= {2 m, nj; (16) 
therefore, dividing = by N, or the sum by the number, we ob- 
tain, as an expression for what you happily call the MEAN VALUE 
_ of the product 2°y2#k, the following : 
: ig um whe 
M dA j2H fe = (20, 2u, 2v} er) to 5 (17) 
or, substituting for { } its value (15), and writing for abridg- 
ment 
K=Atuty, (18) 
: aN oD shes ae SCE 
| eT ee eT Ne aw | (19) 
In oe manner, 
2A+1 neh te yb: ef (2X + Wye (2u) ! (2v) ! 
ae De ee eek (2«+1)! AL pikencarlod eo 
‘The whole theory of what you call the mean values, of 
products of positive and integer powers of ijk, being con- 
tained in the foregoing remarks, let us next apply it to the 
determination of the mean value of a function of w + iw, y + jw, 
_ 2+kw; or, in other words, let us investigate the equivalent for 
your 
Mf (a2+iw, ytjw, zt+kw): (21) 
by developing this function f according to ascending powers 
of w, and by substituting, for every product of powers of 
ijk, its mean value determined as above. Writing, as you 
- propose, 
; d d d 
——— —_ = op 
dw > dx Dy dy 
‘we are to calculate and to sum the general term of (21), 
namely, 
d 
29 — D;, (22) 
R2 
