782 
MR, J. W. L. GLAISHER OR RTCCATI’S 
The six particular integrals of (4) are 
Uo = 1 
= 1 
R,= l 
flVr 
1 2 2 (2 ? -l) 
1 2 ? (29-1)4 ? (49- 
aVj 
1 2 ? (2 2 + l) 
1 29(29 + 1)49(49 + 
Q — 1 
1 \ 
(9 —1)(39—1) 
2(2“ 1) ~ 
h 2(2-1)22(22-1/ 
2 + 1 criA 
(9 + l)(39 + l) 
2(2 +1)"" 
2(2 + 1)22(22 + 1) 
I ^ — ^ r/ ~y 1 
(9 —1)(39- 1) 
2(2-1) 
2(2-1)22(22-1/ 
. 9 + 1 
4- - . c . z ' i - 
, (2+l)(32 + l) 
r , ~ ' 
a 6 + 
G~6 
(V‘Z 
-f&C., 
+&c. L 
(2 + 1 )( 3 ^ + 1 )( 5<7 + 1 ) 
O O/y 
- 
O Ort | 
-f 
(?+!)(% +1)(5?+ 1 ) 
2(2+1) ~ q{q + l)2q(2q + 1) 2(2 +1)%(2?+ 1)32(32 +1) 
-« 3 2 3 ?-j-&c. j e« , 
^a 3 2 3? +&c.|e 
1 - a & 
-cch Zrj -\- &c. i e f~. 
The differential equation admits of integration in a finite form if q = the reciprocal 
of an uneven integer, and, the terminated series being denoted by accented letters 
as before, the relations between the particular integrals are the same as in art. 5, viz. 
(1°) q not = the reciprocal of an uneven integer, 
P 2 =R,=U 2 , Q 2 =S 2 =V 2 ; 
(2 ri ) q — the reciprocal of an uneven positive integer, 
P.,=E i =U 2 =!(P/+R J '), Q 8 =S 3 =V s =dy » + (R/-P/); 
\2 / 42a 
(3 n ) q = the reciprocal of an uneven negative integer, 
P ; =R, = U 2 =(iyi +(S/-Q/), Qj = S £ =V,= J(Q,'+S s '); 
where 
9 r 
(-D*K) J 
Cls. 
1 2 .3 2 .5 2 . 
The integrals P 2 , Q 2 , P 2 , S 2 were given by Cayley in the ‘ Philosophical Magazine/ 
Fourth series, vol. 30, pp. 348-351 (November, 1868). 
