BEAM OF KECTANGULAIl CROSS-SECTION UNDER ANY SYSTEM OF LOAD. 113 
Now 
pn-¥r 
1 
+X+,.. + a;«'- + — 
Difierentiate n times with regard to x. 
1 
= 1 + (n + l)x + . . . d- 
r(r + 1) — 1) 1 (D / x’’-'''’’ \ 
n I 
-j- 
n ! dx”' \ 1 — X 
(1 - 
The remainder is therefore 
1 d'^ 
n\ dif (1 — a;) ~ (1 — ‘ ‘ 
I 0 ^ + r — 1 ). , . + /• — s 4 - 1 ) x’‘+>-^ ('ll + ■;■)...(/■ + 1 ) z’- 
s! (1 — + • • • + , 1 — a; 
This holds for all values of x however near to 1. Putting x = e~^“ omd substi¬ 
tuting in J„, we find J„ = 1st r terms of the series -{- a remainder term consisting 
of the sum of (/i -{- 1) integrals of the form 
9 r (lf± - iTj) g-(o« + n-n + 3)« (^^OMsinh ^6+»coslm0 
► 0 
S ; 
(1 — 
„ ’ !■ r. j 1,1 1 , (a + r) (n + r — 1 )... (n + r — s 1 ) , . 
s ranging from 0 to ii, and the product ^-lieing 
replaced by unity for s — 0, 
Now sinh u is always < u cosh u ; hence the general integral in tlie remainder 
(the factor multiplying cos vz in the integrand being positive througliout) is less 
than 
(n + T)(n + r - 1). . .(n + r - .s + 1) , , , ,, , , , , ,, , / 4« \« + i-^ , 
- {iuy t + cosh W (-yr^-J cln, 
v'O 
i.e., than 
1 - e- 
p (n + _ 1) ^ + , _ , + 1) ' du. 
i-'o s! ^ ' \siiili2a/ 
TVI 
ow < 1 always, and cosh ic < c!\ The general remainder term is therefore 
less than 
r (n + r) (71 + r ~1). . .( n + r - s + 1) 
Jo 
s! 
( 4 w)" + 
^ ] (' 1 ^ x) (n r 1) . . . (?i + 7’ — .S' + 1) p . ^ 
< 4 --,- , _, --tor s ran sms: irom 1 to n. 
/ ,s 
n + r~ - 1 
C* i=> 
For 6' = 0 the remainder term < 4 
s 1 \ 
Tlius for every value of s the 
n + ■?' — ' — 
4 ; 
VOL. cor. — A. 
