183 
ere 
Al ply! ee) 
[A, Ms v j= 
as, I have no doubt, you had determined it to be. 
«¢ With the same signification of { }, we have, by (2), 
Ura ro Vode ni; (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°2#k”, the following : 
“on {A; Ms v| "4 
Wyp2 fr2v 2Ay2u [r2v « 
M Pp Ke =P, Qn 20] PNP: (17) 
or, substituting for { } its value (15), and writing for abridg- 
ment 
k=A+pty, (18) 
aujov . LS «! (2A)! (Qu)! ee) 
M rye ‘Coa ee (19) 
In like manner, 
t(-1)* «! (2A +1)! (Qu)! (2v) ! 
F2A+1 72, ve 
Me = (2e+1)! A! pe! vl- (20) 
«‘ The whole theory of what you call the mean values, of 
products of positive and integer powers of ijk, beimg con- 
tained in the foregoing remarks, let us next apply it to the 
determination of the mean value of a function of x + ww, y + jw, 
z+kw; or, in other words, let us investigate the equivalent for 
your 
Mf (2+tw, yt+jw, 2+hw): (21) 
by developing this function f according to ascending powers 
of w, and by substituting, for every product of powers of 
wk, its mean value determined as above. Writing, as you” 
propose, 
d d d d 
ee WP» Bye dz 
we are to calculate and to sum the general term of (21), 
namely, 
SB (22) 
R 2 
