BEAM OF EECTANGTJLAR CROSS-SECTION UNDER ANY SYSTEM OF LOAD. 147 
being’ now of the (n 2 )“^ degree, we should have enough constants to he ahle to 
identify Q and S with any two, given polynomials on either face of the beam. 
As a mattei of fact this is not so ; for there are solutions, namely, those for: 
(i.) a uniform longitudinal tension, (ii.) a pure bending couple, (iii.) bending with 
constant shear, which make Q and S zero over both faces and yet do not annul all 
the 4n — 4 free constants. There must therefore be relations between the 4 n — 4 
equations giving the constants. They are not all independent and, consequently, 
not every system of surface stress expressible in polynomials corresponds to a 
solution of this type. 
40. Case of n = i. 
Let us see what surface conditions can be satisfied liy the solutions of the fourth 
order. 
In this case, remembering = — B 4 / 4 , 
Q = (t - C.) + (t - X + (- - 2D,) y 
+ (it* ~ ** + (~ IBs — 6 D 3 ) xy + (IA, + 3 C 3 ) tf 
+ 12A^.x/- -h (- 3B, - l2Df)xhj -h (5B^ + 4.Df)y^ 
^ + '^ 1 ) + ( 2 ^ + + ^ - 203 ) 2 / 
+ + (_ IA 3 _ 6C,)xy -h (- A/B 3 - 3 Do) 2/3 
+ (- 9A^ - 12Cf)xhj - 12B^xy^~ + ( 7 A^, -f- iC^)y\ 
and 
+ 9 
U = I ^0 + + Ao {x- — /) 4 A3 {x^ — Sx 7 j^) 4 A^. {x^^ — 6xy 4 y^) 
Sf, (y + ^) I 4 Bi2/ 4 ^B.xy 4 B3 (3xV - y^) 4 B, { 4 x^y - 4 xy’^) 
4 _11 Bi2/ 4 2B3X2/ 4 3B3 {x^y — 2/3) 4 4 B^ {xhj — 3x2/3) 
1 — 2 A, 2 /^ — GA^xy^ — 4A^. (3x^y- — y'^) 
L 1^0 + Cix 4 C2 (x2 - 2/") + C3 (x3 — 3 x 2/3) 4 (x^ — 6 xY 4 7/) 
V 1 4 l)yj 4 2J).2xy 4 D3 (3x32/—2/3) 4 { 4 :X^y — 4x1/3) 
y = |“® 0 -BiX-B 3 (x 3 - 2 / 3 )-B 3 (x 3 - 3 x 2 / 3 )-BJx'^- 6 x 3 y 34 y 4 ) 
8 fi{x’ + ^) I 4 Ai 2 ,' 42 A 3 xy 4 A 3 ( 3 x 3 y- 2 / 3 ) 4 A,( 4 x 3 y- 4 x 2 / 3 ) 
_ -L + 2 ,A^xy 4 3 A3 {x^y — y^) 4 4 A^ {x^y — 3x1/3) 
4 /^ 1 4 ‘2.B.yf 4 6B3X2/3 4 4B^ { 3 xY — y‘^) 
_p i_ j ^^0 + + D3 (x 3 - 2 / 3 ) 4 D3 (x 3 - 3 X 1 / 3 ) P)^ ^^4 _ Q^ 2 y 2 _p ^1 
2 /. [ - Oil/ - 2 C>y - C3 ( 3 x 3 i/ _ 2/3) _ 0 , ( 4 x 3 ;i/ - 4 xy^) 
u 2 
