77 8 
MR. J. W. L. GLAISHEE ON EICCATI'S 
(- 1 / 1 
(i + 1)! 4 i+ V +1 
«{(?' + !) . . . 2i}2x+j,{i . . . (2i—l)}2.2 2 x 2 . . . 
[1.2 .. . i}(i+ l)2 i+1 x i+l 
fi +1 
(t+ 1 )!' 
Y 2i+2 
(2i + 2)! 
,{(-f)(-f+l) . . . (-l)}(2i+2)2 si+ V‘' +a 
y 2i -t-3 
F(^y ! {(-^- 1 ) • • • (-2)}(2{+3)2^»0^8+& c . 
If i i(f-l) a2a50 
A l+-ax+ -7T ITT • • 
x l |_ i 1 — 21 
i(i —1) . . . {i — (i — 1)} ate* 
• • • {»—$(*—!)} i' 
+(-)'« 
(2t+ 1)! 
a 2 *' +2 P + ffi 1 
f + 1 (i'4 l)(t + 2) A ; 
rax 
f+1 1 (i+l)(i + f) 2! 
o i +&C. 
The coefficient of ti +l in the expansion of e ax . eh ^C* a +*©+*} is therefore 
XR'- 
-,2i + 2 
(i+l).(2i + l)! 
S; 
and we have 
1_ g 3 *+ 3 
\ (i+l).(2i + l)! 
_/ _ 2(2i +1) 2j+i_ o 
{1.3.5 ... (2i+l)Y a ~~ 2g ’ 
so that the coefficient of h i+l 
:\(R'—2g 
14. Thus the three forms of the same integral which are obtained by the expansion of 
^aV(x-+x/i) qCix gc{ */[x*+xli)— .r} g— a.c g«j -Jix^+xlfi+x} 
are 
TJ-gY, P', R / -2 fl rS. 
Changing the sign of a, we obtain as the coefficient of h‘ +} in the expansion of 
g ~aV(x-+x/i) g— ax g—<?{ .?} guz g-s{ Vfc 5 +ii)+i} 
the values — X(U-j-yV), — XIT, — X(P' + 2 gQ), giving the three equal integrals 
U+<7V, B', P' + 2<7Q. 
