383 
in the ‘Cambridge and Dublin Mathematical Journal,’ Jan. 
1847, and was shown to represent the most complete solution 
of the equation, with reference to the problem of the attrac- 
tion of a solid of revolution on an external point.* See also a 
memoir by Hoppe, on the problem of the motion of conoidal 
bodies through an incompressible fluid (‘ London Quarterly 
Journal of Mathematical Science,’ No. IV.). In its general 
form, i. e. antecedently to the determination of the function ¢, 
this integral cannot be freed from imaginary quantities with- 
out development. When, however, the function ¢ is known, 
it may be freed from them without development. It appears 
to me, therefore, that the solution possesses, im respect of its 
relation to imaginary quantities, an advantage over the form 
which you have deduced by the method of quaternions. 
‘‘ The direct symbolical solution of the equation, 
Pu du du 
de * dy* de = 0, 
obtained by resolving the operating ae into factors of the 
d ef @\ a d? 
form ant (zt a) 7-1, and — - -($+ =) v7 -l, is 
a? a? a nk 
u = COS fo s+ ) lA (y, z) + sin fe of a+ m) {yf (y, 2). 
If, in consequence of the arbitrary form of the functions 
Si(y, 2), f2(y, 2), we replace the latter function’ by 
d d? \3 
(= 47 za) g2 (y; Zz), 
wherein ¢, (y, 2) denotes also an arbitrary function of y and 
z, and then for symmetry write ¢, for {, we have 
u= cos 2 nisi im Pi Z # + alias 
ae dye * dz? pi (Ys. ) aE sin te dy? dz 
ad? a 
dys * dz $2 (Y> 2), 
* I have since obtained, in the form of a double integral, the complete solution 
of the equation under the condition referred to. 
