~ 990 
the solution of Laplace’s equation were presented in the paper 
of February, 1852,—namely, 
a f SAem224 | cos 4/ (m? + m3) z+%,. sin / (mi +m) z} 
= 4 + 
[ = Bem" {cos (mi + m2) z-%,. sin / (mi +m) z} | 
with its duplicate 
VAeinm2 | cos (myx + my) + tp . sin (myx + my)} ) 
+ iy 
= Bem») (cos (mx + my) — t, . sin (mx + my)} | 
where 
2 ¢=Ccosat+jsina; 
and 
[J® (7m, m.) em . cos +/ (m? + m3) 2. a 
+ 
L {J (am, mz) em". sin / (m? + m3) z .dm,dm, 
with its duplicate 
(m1, Mz) Cos (me + may) e™*"2)2 , dm,dm, ) 
y 
= + 5 
W (m,, mz) sin (1,2 + my) eMn**m2")2 , dm,dm, 
Y J 
the limits of the integrals in both cases being supposed inde- 
pendent of the quantities x, y, and z,—stand unaffected.” 
Professor Graves observed, with reference to Mr. Carmi- 
chael’s paper, that he entertained great hopes that Mr. Carmi- 
chael would succeed in discovering the requisite modification 
of the symmetrical expression now exhibited by him to the 
Academy, so as to make it actually satisfy Laplace’s equation. 
Professor Graves stated that he had pursued the same track 
of investigation himself; but he had abandoned it in conse- 
quence of his finding that the expression 
MI 
m Seman are} 
in which W denotes an arbitrary function, is not a true solu- 
tion of Laplace’s equation. This becomes at once apparent 
on trying the case in which the function just given reduces to 
2 
] ; 
