a Se ae 
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 + je, z+ kx) + Mf, (y —ja, z- he), 
which I lately presented to the Academy as the complete so- 
lution of the equation 
Vi BV Gav 
area ay tes = 
‘¢ 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+jc, z2+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 
F,+jP,+ kFs, 
af P\ a/df . af df 
eT si(qat +h) +Fil gat? aeet 2)- oor 
ae a af af of 
aa -5i( gs ala at pag aa) < 
df #/( af adf\ #/ df af def 
Br ae a gee ligase nees a3) Bee 
where 
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 
ae is, 
(-1)"f @& @ |" 
(On)! jaye def 7 
In F, the coefficient of 22" i 
X-1)ry 1)" a @& | n j 
ee ie cdet ts 
