780 
MR. J. W. L. GLAISHER OX RICCATI’S 
This equation therefore admits of integration in a finite form when p— an integer, 
and the six particular integrals V,, P x , Q 1? lt L , S ls winch are equal respectively to 
x p \ J, x p Y, x?P, x p Q, x'Px. x^S are connected with one another, in the different cases, 
by the same relations as those found for U, Y, P, Q, P, S in art. 5. 
If we put 2 p=n —1, so that the differential equation becomes 
d~v 
fa? 
n —1 civ 
x dx 
a?v=Q 
(3), 
then the six integrals take the forms 
o 
U 1= 1 
1 9 c 
fax' 
W4 
fra 
- _ 9 9 
(n—2)(n— 4) 2 2 .21 (n— 2)(n — 4)(?i — 6) 2 3 .3! 
Y x =x n J l 
1 o cj 
arx 1 
n-h 2 2 (?i + 2) (a + 4) 
92 9 
(» + 2)(ra + 4)(» + 6) 213! 
P — J 1 (rc-l)(w-3) ah? (n-l)(n-3)(n-5) ah? , 
1 ' rc-1 i "(«—l)(7i-2) 2! (»-l)(»-2)(n-3) 31^ *> ’ 
q =£C « J i _d± 1 r , r i -t!)(«- +3) «¥ (ra + l)(?z. + 3)(w + 5) aV & , 
' n +1 + (w+l)(?i + 2) 2! (n + l)(n + 2)(n + 3) 3! ' ' ’ 
R i= ^ i+5=i a» ~+&c. 
(w-l)(»-2) 2! 1 (n-l)(n-2)(n-3) 3! 
O „ I, . n + 1 (« + l)(Y + S) crfa (?i+l)(% + 3)(?i + 5) a?x % , 0 , 
n- )TTI <» + ( - t+1)( , 1+2) -5r+ (, + i ) (, +2) (. + , ) it + &0 - ^" • 
The differential equation admits of integration in a finite form if n— an uneven 
integer, and the relations between the particular integrals are the same as in art. 5, 
viz., accented letters denoting the terminated series as before, 
(1°.) n not = an integer, 
P 1 = R 1 =U 1 , Qi = Sj = Vi ; 
(2°.) n— a positive uneven integer, 
Q 1 =s 1 =v 1 =E(R ; '-P l '); 
P 1 =R 1 =U 1 =i(P 1 '+R I '), 
