812 
MR. J. W. L. GLAISHER OR RICCATI’S 
and 
are respectively 
and 
d 2 y 2 i dy , 
s +^ =0 
1J—X- 
1 
'1 
(1 
- 
— . . 
V 
X 
X 
\x 
1 
“1 
(l 
/ 
V 
X 
X 
\x 
where v—c sin x x /n-\-c 1 cos x^/n, and the former of the two expressions involves i 
differentiations and the latter i-\-1 . 
In the ‘Philosophical Magazine’* for May, 1856, Mr. Benjamin Williamson 
obtained by a symbolic method the integrals of the differential equations 
d s -“d+A=o, (D-+ ?S 7 1 ) D+« 2 )y=o 
X 
in the respective forms 
d 
y= A^—a 1 j cos (ax -f- a), y— Ax ' 2i 
and that of the equation 
-2i + l _ 1 
\da 
a 
cos (cac+a), 
+A =0 
in the form 
y=Ax ‘ 
cos (ax-\-a ); 
and in the ‘ Philosophical Transactions’! for 1857 the late Professor Donkin obtained, 
also by a symbolic method, the integral of this last equation in the form 
y—x { D - ) (c x sin ax-\-c % cos ax). 
* “On the Solution of Certain Differential Equations” (‘Philosophical Magazine,’ Fourth series, 
vol. xi., pp. 364-371). 
f “ On the Equation of Laplace’s Functions, &c.,” vol. 147, p. 44. A proof that the integral of the 
partial differential equation — =-=P + - - " — ^ 1 ^ u, which is a simple transformation of (1), may he 
a~ at* dr 1 r dr r 2 
presented in the form u =r l ^' ~f ( '0 + y(' —«0 £ g g{ ven py Professor C. Riven in the ‘ Solutions 
of the Senate-House Problems and Riders ’ for 1878, pp. 158, 159. 
