193 
et iDakDs) if (y, 2), 
is to be interpreted as follows :— 
“ Substitute y+ jz for y, and z+ ke forzin f(y, 2); taking 
care to leave all powers of j and & in evidence, and then re- 
place all the products of js and 4s, obtained in this way by 
mean products of those imaginaries. 
‘¢ Reasoning and processes in all respects similar lead to 
the conclusion that the effect of the symbol 
Pe +i DytkD3) 
upon any function whatsoever of x, y, and z will be to change 
it into the same function, in its mean state, of w+ iw, y+ jw, 
and z+kw. By this it is to be understood, that after this 
change of the variables has been made, and the development 
effected as if 7, 7, and k were ordinary algebraic quantities, 
mean values of products of the imaginaries are to be substi- 
tuted for ordinary ones. 
‘«¢ Reverting now to the solutions of the differential equa- 
tions noticed in the first pait of this Paper, p. 168, we see 
that they hold good. without any liniiation of the nature of 
the arbitrary functions, provided we mocify, or rather perfect, 
our conception of the mean sia e of a function in the manner 
just described. 
“¢ Our new definition of a mean product, ov of a mean func- 
tion, coincides with thai given ep. 166, in the case where m 
and m are positive integers: and it includes the cases where m 
and z are negative or fractional, to which the original defini- 
tion of a mean product is inapplicable. 
‘* If it should prove that the solution of Laplace’s equation 
now attained to, viz. : 
V= Mf, (y + ja, z+ kx) + Mf, (y -ja, z-kz), 
is something more than a mathematical curiosity, and answers 
the demands of physical inquiry, we shall have reason to re- 
joice not only in the fruits of that particular discovery, but 
