806 
MR. J. W. L. GLAISHER ON RICCATI’S 
dz & 
=^— (z*i +i -Y —— 
3»J+2?y dz) £k~H* 
o 4-i d V +1 -« 1 / o -i-i d\ i+1 ei 3 
r 2 ? +l ) g 2 = o~4 J z S?+1 T — ; 
dz) z^ + *i\ dz) z % i 
so that the solution of Riccati’s equation may be written also in the forms 
u— - f z 2?+1 
2 # 
u 
=4^ 
AY 
(V* 
+ c 2 e « 
dz) 
2 l-3? 
a q 
a 9 
l1V' +1 
( c 
+ c 2 e s' 
dJ 
V 
dV 
if q=^ +1 , and 
w=z 1_ ? ) {z 3 ? +1 (e 1 ei 3 fi-Coe - ^)}, 
w=z 1-3 ?(z 2 ? +1 ^j {z? + i (c 1 e4' -j-c 3 c s')}, 
lf ^ =_ 2i + l* 
35. Boole’s form (29) of the solution of the equation (l) can be obtained also from 
the definite integral (18) in art. 29 ; for we have 
, — x y+\[ 
J i 
U- 
COS at; 
>o(aP + ?*)* + 1 
d(=x-r-4 , if b=x~\ 
J n 
(i +i?y 
v 
i 
X 1 1 
=i*-wr*»Y i\ o 7 t~kM 
a+i'i-y 
00 \ 0 
V- 
= 1 tt . - x -/’- 1 
P- 
da] 'ii 
i y—T '%dg, when p is a positive integer, 
J r, 1 + ot;~ 
p e Vb 
~7b 
