54 MR. HAIIGREAVE ON THE SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS. 
the symbol denoting that, after the operations are performed, ?/—2c<r must be 
L 
written in lieu of?/; and the symbol denoting that, after the further operations 
are performed, y-\-cx must be written in lieu of y. 
The solution is simplified by considering the latter function alone as a partial solution, 
and completing the solution by changing the sign of c with a new arbitrary function. 
Now if in Laplace’s equation 
we make x=tan~*(|W-\/— 1), 
we obtain 
d^u d'^u 
dx^ dy^ 
—n{n-\-\) 
u 
X 
=0. 
The solution of Laplace’s equation, therefore, by this process assumes the form 
tan-‘(f/-^/^))}. 
Postscript. — Received March 16, 1848. 
The following brief investigation is more general in its results than that developed 
in pages 50 and 51. 
By applying the fundamental theorem to the linear equation 
U^+„—<pX.U^='4yX, 
and its solution 
u^=^(x—n),(p(x — 2n)...'X{(ipx.(p(x—n).(p(x—2?z)...)~^\px}, 
(where the sign of summation has reference to x, Ax being w,) we obtain the equation 
s~’^u — <p(D)u=^px, ( 12 .) 
and its symbolical solution 
m=(P(D— w).(p(D— 2 w)...|^ 3^:^((<P(D).<P(D— n).(p(D-2w)...)-‘-4/x)| ; 
and by expanding the factor ? and reducing, we obtain for the solution of (12.) 
the series (2 referring to p), 
?«=2e“^”"’(^(D).<p(D — n)....(p(D — pn))~^'^px, . 
= 2(9(D-}-pn).(p(D+(jo— l)n)...,(p(D))~'^{s~^”'^'^x}. 
If i-=-y, and %y consist of powers of y, the above formula gives the solution in 
series of powers of ?/ of the equation 
Several equations of this form solved by Mr. Boole’s general method, are given in 
the Philosophical Transactions for 1844, pp. 236-240. 
