Prof. Donkin on the Equation of Liaplace’s Functions. 65 
to the preceding estimate, could be distinctly conveyed by a sub- 
merged wire of 500 miles in length, could of course be easily per- 
formed by the hand, with the aid of a key-board and clockwork 
- power adapted to make the double operations for giving rapid sub- 
sidence of electricity in the wire when any one key is touched, and 
to let the different strengths of current, in one direction or the other, 
be produced by the different keys. Thus without a condensed code, 
thirty words per minute could be telegraphed through subterranean 
or submarine lines of 500 miles; and from thirty to fifty or sixty 
words per minute through such lines, of lengths of from 500 miles to 
100 miles. 
The rate of from fifty to sixty words per minute could be attained 
through almost any length of air line, were it not for the defects of 
insulation to which such lines are exposed. If the imperfection of 
the insulation remained constant, or only varied slowly from day to 
day with the humidity of the atmosphere, the method I have indi- 
cated might probably, with suitable adjustments, be made successful ; 
and I think it possible that it may be found to answer for air lines of 
hundreds of miles’ length. But in a short air line, the strengths of 
the currents received, at one extremity, from graduated operations 
performed at the other, might suddenly, in the middle of a message, 
become so muca changed as to throw all the indications into con- 
fusion, in consequence of a shower of rain, or a trickling of water 
along a spider’s web. 
* On the Equation of Laplace’s Functions,” &e. By W. F. Don- 
kin, M.A., F.R.S., F.R.A.S., Savilian Professor of Astronomy, 
Oxford. 
The equation ihe Bn 
q+ iy st as =0, when transformed by putting 
z=r sin 6 cos¢, y=rsin@ sin ¢, z=rcos@, may be written in the 
form 
Dee aN. cc Cte 4 
{ (sin945) +(3) + (sin 6)’r— ft 1) bu=o; (1) 
and if u=uy4+ur+u r+... +U,r"+..., we find on substituting this 
value in (1), and equating to zero the coefficient of 7”, that w,, satisfies 
the equation 
{ (sino 5 + (J) treet Ging } woo, = Fe cish2) 
commonly called the equation of Laplace’s functions. If we put 
sin 0 +n cos 0=q,, then the equation (2)*may be written 
(m7 +77 + (%) = 0; 
dy 
and the operation @, possesses the following property, namely 
Dann +N =Dp—1B-(n-1),+(n—1)*; 
hence it is easily shown, that in general the complete solution of (2) is 
Un= Dn Dn—1 +++ WW) « Upy 
Phil, Mag. 8. 4, Vol, 14, No, 90, July 1857. £ 
