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IV. On the Equation of Laplace’s Functions, &c. By W. F. Donkin, M.A., F.RB., 
F.B.A.S., Savilian Professor of Astronomy in the University of Oxford. 
Eeceived December 3, — Eead December 11, 1856. 
Section I. 
The equation 
Pu d'^u d% „ 
dx^ dy'^ " 4 ” dz"^ ’ 
transformed to polar coordinates by putting 
a:=f sin ^ cos <p, ^=rsin^sin^, zz=r cos 6, 
becomes, as is well known. 
d'^u , , Au 
^+COt: 
^^4.r2^4-2r--0 
*2+' ^r- — 
dr' 
</9 ~ (sin 9)^ d(f’ 
which may be written in the form 
(sin 4) +(^) «)'4r(4+l)}“=“ ’ 
( 1 -) 
and if it be assumed that 
u=Uf,-{‘UTr-\-u^r^-\- ? 
then by substituting this value of u in either of the two equations last written, we find 
that is a function of ^ and <p satisfying the equation 
+(^) +w(w+lXsm^)^|%„=0, (2.) 
which, under a slightly different form, is commonly called the Equation of Laplace’s 
Functions. 
This equation was first solved in finite terms by Mr. Hargeeave *, but in a form very 
inconvenient for practical applications. A solution free from this objection was after- 
wards obtained by Professor Boole j’, by a method explained in his memoir “On a 
General Method in Analysis,” Philosophical Transactions, 1844. Lastly, in the second 
volume of the J oumal referred to, the same mathematician gave two more solutions, one 
of which however is reducible, as he states, to Mr. Hargreave’s form ; the other, though 
much more convenient than this, is still for most purposes probably less useful than that 
given in the first volume, from which it differs essentially in form, as well as in the 
method by which it is deduced. 
* Philosophical Transactions, 1841. 
t Cambridge and Dublin Mathematical Jom-nal, vol. i. p. 10. 
