169 
(D+b)tne" Dig, 
(D BA Eye = ewe De ewe. 
Consequently, 
V= er? D2 ¢2ure D2 eur U 
+ refi (2; ys Z) 1 a’? fa (a, Ys Z)s (7) 
and to interpret this we must ascertain what is the effect of 
the symbol 
Ww D> 
G's 
upon any function of x, y, and z. 
‘* Reasoning as before, we should find that its effect upon 
a term 2! y™z", 1, m, and n being positive integers, will be to 
change it into the 
‘Sear part of the sum of all the a 
differently arranged products of the 7+m-+n factors, of which 
Jare equal to x+iw, m to y+jw, and nto z+kw. In other 
words, to change it into the mean value of this product, that 
is, into 
M (2+iw, y + jv, z+ hw): 
and, more generally, the effect of 
(pile 
upon f(x, y, z), any function consisting of positive and integer 
powers of w, y, and z, will be to change it into 
Mf (a+ iw, y+jw, z+ kw). 
We are thus enabled to interpret the formula (7), when U, 
JA» and f;, are functions of positive and integer powers. To 
that case the demonstrations given in this paper are essentially 
limited. I hope to be able to lay before the Academy the 
investigation of the cases where /, m, and n, are negative or 
fractional. So far as I have yet discussed them, they seem 
to present results which it is more difficult to express with 
elegance. ; 
‘In conclusion, I may be permitted to state some theorems 
at which I arrived whilst discussing the subject of the present 
