186 
The Rev. Professor Graves, D.D., read the second part of 
his Paper on the solution of the equation of Laplace’s func- 
tions. 
‘In the former part of this Paper I showed that the sym- 
bol 
(7 Dy+kD. 
é J+ 2, 
or 7, as we have denoted it for the sake of brevity, when ope- 
rating upon y” 2”, has the effect of changing it, if m and x be 
positive integers, into the 
((m+n)!\* 
m!n! 
(m+n)! 
m!n! 
part of the sum of all the 
differently arranged products, of which each contains m factors 
equal to y+ jx, and equal to z+’. But I reserved the con- 
sideration of the cases in which m and n were negative or frac- 
tional. In fact, I had ascertained by trial that the theorem 
just announced must undergo some modification in its state- 
ment before it could be extended to the case where m or n was 
negative; and I was at a loss to conceive what modification 
could render it applicable in the case where either of the ex- 
ponents was fractional: the rule given for the formation of a 
mean product seeming of necessity to presume that the expo- 
nents were at least integer, if not positive numbers. In the 
present communication I desire to lay before the Academy the 
discussion of the reserved cases. In dealing with them I have 
been led so to modify my definition of a mean product as to 
make it apply where m and n are negative or fractional; at 
the same time that it coincides with my previous definition 
in the case where m and » are positive integers: and this 
has been accomplished by the help of mean products of 7s, 
js, and ks, the fundamental theorems respecting which were 
stated at the end of my former Paper, p.170. Thus it will be 
found that we are in possession of a complete and perfectly 
simple solution of the equation of Laplace’s functions :—com- 
plete, as involving two arbitrary functions ; and simple, as it is 
