148 
MR. L. X. G. FILOX OX AX APPROXIMATE SOLITTIOX FOR BEXlMXG A 
P = 
^ + Cl) + (-■ + 20,)+ y 
oB., 
4- 2D< 
+ + 30,) + .T,/ (J-iB, + GD.) + / (- - 3Cy 
+ a- (3A, + 4C,) + xy/ (loB, + 121),) + xf (- 2lA, - 120^) 
+ ?/'^ (— — 41) 
and we notice that, in virtue of the relation B^, = — 4D^ the coefficient of x-y in Q 
goes ont. Hence the coefficient of x~ is the same for-Q^^ and Q_i. This alone 
shows that the solution is not the most general that can he got, given that the 
stresses on the np])er and lower surfaces are (|nadratic functions of x. 
§ 41. Determination o f the Constants for a Beam Umfio-mly Loaded. 
Here we have, over tlie upper surface y = -{- h : Q = constant = q say ; 
over y = — b : Q = 0 ; and over 2/ = i : S = 0. 
The last two conditions imply 
Bi + 41), + V ('- _ 31)^) = (,.(Ill), 
+ 2L); - 12B,GS = 0.(112). 
-- 20, + Gy-A., + 4C,) = 0.(113). 
--IA3-CC, = 0.(114). 
2S‘ + 3D3 = 0.(115). 
-9A, - 12 c, = 0.(IIG). 
(IIG) and 
40, = A,. (117), 
give at once 
A, = 0, C,=-().(118). 
The conditions for Q give 
; - Ci-('h + 3D3)5 + ;.-)^“+.3C3) + P(5B, + 4D,) = 7 . (119). 
; - c, + (h + 2 ir+ I'Cf + sc) - ''MSB. + 4D,) = 0 . ( 120 ). 
- 183-0113 = 0 .( 121 ). 
T - 20. + 5ni2A,) = 0.(122). 
SM- 303=0 . (123). 
