ON THE STRAINS IN THE INTERIOR OF BEAMS. 83 
upwards, is —yéx. And the reaction R of a support may act upwards at 
distance h. 
Expanding the sines and cosines, putting éw for sin 6 . ds, and dy for cos 0 « és ; 
putting also 
L=B.sin? 6+C. cos? 6, . 
M=(B—C) .sin (. cos B, 
Q=—B. cos? B—C. sin’ }, 
O=y—Q, 
ma] 
= 
rns the equations of equilibrium in the usual way, they will be found 
to be— 
Equation for forces in 2, [de .(Ip+M)=0. 
Equation for forces in y, [dep +0)—R=0. 
Equation of momenta, fdxCyp +My+ Mep +0xv)—RA=D. 
Now these equations, applying to any curve, will apply to any two curves 
very close together; and therefore their variation, taken by the rules of the 
Calculus of Variations, will be 0. The proper equation (in the usual nota- 
tion) is y—“D)=0. Applying this, the results are 
dM dL _o 
dy da’ 
dO dM _ 
dy da ‘ 
_ From this it follows that (omitting some arbitrary functions which represent 
original strains in the formation of the beam) L, M, O, are partial differential 
coefficients of the same function of w and y, which we may call F; so that 
ra yn @! ont 
dy” dxdy’ da? 
Substituting these, the equations become 
f': “(s, a S.Az Rao! f.d(yF+2-F)—M= 0. 
Considerations, of a somewhat detailed character, depending partly on the 
relation assumed to exist between tension-force and material extension, are 
necessary to show the form which must be assumed for F in the various cases 
to be examined. The conditions to be secured are—that the horizontal part 
of the thrust, &c. shall be the same as that given by ordinary theories, on the 
relation just mentioned ; and that the equations above shall be satisfied. After 
due application of these in the following five cases, these forms are found 
for F. 
Case 1. A beam of CoM r and depth s projeuns from a wall; 
Case 2. A beam of aoe _ and depth s aia at both ends ; 
FHS. (0*—2re). (es —). 
G 
