768 
MR, J. W. L. GLAISHER ON RICCATI’S 
Taking the first root, the equations are 
( - 2 i ? ) A i + 2 ( ~P) a A o=0, 
2(1 — 2p) A 2 +2(1 —j))aA 1 =0, 
3(2 — 2p)A 3 -}-2(2 p)aA 2 =0, 
giving 
Ai= 
«A 0 , 
A 2 = ——juAj, 
P~2 
Ao= ——ruA.-,. 
— 9 
3 ,, 
P~1 
and we obtain the particular integral 
,-jj i_4 r , l>LPrll ah?_ p(p-l)(p- 2 ) ah? 
p p(p-i) 2! p{p—\){p~l) 3! 
Similarly, the other particular integral is found to be 
3CP +1 
1 , i j + l , (p + l)(jj + 2) a~x* (p + l)(p + 2)(p + 3) bW , i 
+ l ^(p-nXp+i) 2l^(p + l)(p+IXp + 2) 3! ‘ 
If we had transformed (1) by assuming u=e~ ax v, we should have obtained a differ¬ 
ential equation in v differing from that given above only in having the sign of a 
changed : and the two particular integrals would differ from those just written only in 
having the signs of the alternate terms negative. 
3. Thus, of the differential equation 
clH 0 p(p +1) 
— — aru= -yr u , 
clx" x A 
we have obtained the six particular integrals U, V, P, Q, II, S, where 
TJ=x-s \ 1 ' 
y = x p +1 - 
1 0 0 
crx~ 
p- 
1 92 
+ 
4,,4 
aw 
(p-i)(p- f) 2*2! (p-i)(p-i)(p-i) 2 6 .3! 
3T++&C- , 
~P +1 2 2 + 
1 
1 
P=x J> -I 1 — — ax 
P 
0+v)0+f) 2*2! O + DO + vlO+v) 213! 
p(p — 1) ah? p(p — \){p— 2) cih? 
Q =zx? +l 
p + 1 
1- —ax 
P(p-\) 2! P(P~A)(P- 1) 3! 
0 + 1)0+ 2) a~x~ (p + l){p + 2)(p + o) ah; 
p + 1 
It — x ? i 1 -j— cix — 
L P 
S=x^ +1 j l-\-^—~ax 
(p+l)(p + |) 21 (p +1)0+ 1)0 + 2) 3! 
p(p — l) a 2 x 2 iP(P —1)0 —2) ah? 
P{P~\) 2! PO-^O- 1 ) 3: 
, ,+&c. 
j 
+&c. je**, 
f + &c. | e ax ) 
+&c. 
(p+1)0 + 2) aV 0+ l)(p + 2)0 + 3) cih? 
p + 1 
0 +1)0+1) 21 0 + l)0 + f)0 + 2 ) 3! 
+&c. \e 
