BEAM OF EECTANGULAE CEOSS-SECTION UNDEE ANY SYSTEM OF LOAD. 71 
= (i i - 20t) 2^ zSy) f F) + X (’?)]- £ { G. (f) + B (>))]. (24), 
S = ! - + x(’/)S+M;f;{G,(f) + F,(,)( . (25), 
dy 
which have the advantage of not containing imaginaries if (/> (^) + ^ (-q), (^) + (rj) 
are real. 
§ 5. Solution involving Hyperholic and Circular Functions. 
Assume now for the arbitrary functions the following typical forms 
(^) = A sin cos -j- E cos + iF sin ?n^, 
x{l) = A sin mg — iB cos mg + E cos mg — I'F sin mg , 
(^) = C sin m^ + t’D cos m^ + G cos m^ -f" sin m ^, 
El ( 1 ?) = C sin mg — tD cos mg + G cos mg — tH sin mg, 
so that 
<^ (^) + X {g) = 2 sin mx (A cosh my + B sinh my) 
+ 2 cos mx (E cosh my —• F sinh my), 
(f) (i) ~ X (■’?) — cos mx (A sinh my + B cosh my) 
— 2i sin mx (E sinh my — F cosh my). 
Gi (^) + [g) =. 2 sin mx (C cosh my + D sinh my) 
+ 2 cos mx (G cosh my — H sinh my), 
Gi (1^) — F]^ (jj) — 2i cos mx (C sinh my + D cosh my) 
— 2i sin mx (G sinh my — H cosh my). 
Whence from (23), (24), (25) we get after some reductions 
(3A' + C') cosh my + (3B' + D') sin my 
+ 2my (A' sinh my + B' cosh my) 
— (3E' + G') cosh my + (3F' + H') sinh my 
+ '2my ( — E' sinh my + F' cosh my) 
P = cos mx 
+ sin mx < 
Q = 
cos mx 
(A' — C') cosh my + (B' — T>') sinh my | 
— '2my [A! sinh my + B' cosh my) 
-p sin.ma; 
r 
(E' — G') cosh my 4- (F' — H') sinh my 
— 'Imy ( — E' sinh my + F' cosh my) 
(26). 
(27), 
