167 
gr DarhDs) t (Ys z), 
in which f(y, z) is supposed to be of the form 
S Aya, 
m and n, as before, being positive integers. 
‘«< The exponental symbol being distributive in its nature, 
~ we shall have the proposed expression equal to the sum of the 
mean values of the products corresponding to the several 
terms such as Ay™2". Consequently, 
gIPwkDs Fy, zZ)= > AM (y+ j2, 2 i kz), 
and, with an interpretation suggested by what has been already 
said, we may write finally, 
| Er IPakDd F(y, 2) = Mf (y+ jx, 2+ ka). 
‘* The boundaries of algebra having been of late extended 
so as to include symbols which are not commutative with each 
other, it becomes absolutely necessary to have the means of 
denoting certain standard and constantly occurring combina- 
tions in brief and unambiguous ways. The symbol M, pro- 
posed in this paper, may perhaps be a useful contribution to 
mathematical language. It has the recommendation of having 
been already used in a similar, though less extensive, meaning 
by M. Cauchy. It may also be regarded as an extension of 
Sir William Hamilton’s notation of § (a, 3), which stands in 
the Calculus of Quaternions for $ (a + Ba). 
«« Knowing how to interpret the expression, 
aj DztkD. 
we are enabled, in general, to solve the equation, 
DiV + DjV + DV = U, (6) 
in which U denotes any function expressed by means of posi- 
tive and integer powers of 2, y, and z. The solution depends 
upon our being able to invert the operations denoted by 
D, +jD,+ kD;, and D, ~jD, —kD,; 3 
