84 ; REPORT—1862. 
Case 3. A beam like the last, carrying a weight W at the distance a from 
one end. 
In this case the function is discontinuous; its forms are— 
"ea (VW). 
je—a*} .(f - 
From «=a to «=2r, F=S : {-N2+(2—w2)o—a* } , (4) : 
From «=0 tow=a, F= o {(4+W=S 
s Ts 
s 
(Of this case, two instances are given in the curves below.) 
Case 4. A beam like that in Case 2, with a straining momentum applied at 
each end, as in the middle tubes of the Britannia Bridge ; 
poe (ee r) 
ee _ 6 . 
s? 
Case 5. A beam like that in Case 2, with a straining momentum applied at 
one end only, as in the exterior tubes of the Britannia Bridge ; 
re 
ns pein |e 
s 4 6 
By forming the differential coefficients of F symbolically, L, M, and Q 
(=y—O) are obtained in a form which admits of numerical calculation for 
every value of « and y. And from these, B,C, and 6 are computed without 
difficulty. 
In this way the values of B, C, and 6 have been found for every combi- 
nation of the values v=r x01, e=rx 0:2, v=rx 0-3, &e., with the values 
y=sxO01, y=sx0:2, y=sx0°3, &e. In Case 1, 121 points were thus 
treated: in each of the other cases the computations were made for 231 points. 
Tn the following diagrams are given the curves representing the directions 
of pressure and tension through the beam, together with a few numerical 
values at the most critical points, for each of the cases to which allusion has 
been made. 
CURVES REPRESENTING THE STRAINS IN BEAMS, UNDER DIFFERENT CIRCUMSTANCES, 
The continuous curves indicate the direction of thrust or compression; the 
interrupted curves or chain lines indicate the direction of pull or tension. 
The figures denote the measure of the strain; the sign + meaning compres- 
sion, and — meaning tension. The unit of strain is the weight of ma- 
terial lamina whose length = depth of beam. 
y No. 1. Beam projecting from a wall. 
