221 
_ Professor Graves, D.D., read a Paper on the solution of 
the equation of Laplace’s functions. 
‘It is not my design, in the present communication, to dis- 
cuss the results obtained by giving particular forms to the ar- 
bitrary functions f, and f,, which enter into the expression, 
V= Mf (y+ ja, 2+ kx) + Mf, (y—ja, z- ke), 
which I lately presented to the Academy as the complete so- 
lution of the equation 
EV eV dV 
ia) wy ae 
‘¢ But I propose to give some development to the general 
formula, in order more plainly to exhibit its nature and the 
mutual relation of its parts. For this purpose let us take 
Mf (y + jx, 2+kx), and after developing it by Taylor’s The- 
orem, let us substitute mean products of js, and 4s for the or- 
dinary products, according to the formule of p.170. It will 
then assume the form 
P+ jh, + kFs, 
Cf aN. a4 ask d'f eel 
ae ni(qet we) a & 4 2 ade a) oo 
B= ie Hla a) silent? ae sail ue) Ke. 
where 
“ly 3!\dy? dydz*)  5!\ dy? ~ dysdz? * dydz 
ee Bape GEN ama fe af sf 
#y= eae a ° 
da 8 51 apa ; 7) ws (ayia: 2 apde 7) i 
J being used for brevity to denote f(y, z). It is very easy to 
ascertain the law according to which the coefficients of the 
different powers of x are formed. In F, the coefficient of 
(- 1)" { d? @& n 
(Qn) lay az 7 
In F, the coefficient of 2?"*} is, 
Glial, sf d2s ye 
Qn+Ildy [ap def 7? 
a is, 
