Mathematical Discussion. 
25 
other two (produced if necessary), and let F be the projection 
of F' upon A B. 
Let l — the span A B; 
h = AF; l 2 = F B; 
n = panel length C D; 
n x — C F; 7i 2 — FD; 
Pj = total load on A C; 
P 2 — total load on D B ; 
Q = total load on C D. 
Then when the required stress is 
the following condition is satisfied: 
n i 7i/ o 
*. + «'« *.+ «*« 
a maximum or a minimum, 
•( 1 ) 
Under this general rule there are three principal cases to be 
considered, according as the point F falls between C and D 
(Fig. 2), between A and B but outside the length C D (Fig. 3), 
or outside the span A B (Fig. 1). (In the figures the member 
whose stress is under consideration is in each case marked H K.) 
In these three cases some or all of the quantities n v n 2f l x and l 2 
will have different signs. To determine these signs in any case 
let distances from left to right be regarded as plus, and distances 
from right to left as minus; then the values above given for 
l x , l 2 , n x and n 2 are correct in sign for all cases. 
If the loads are concentrated, the general condition (1) re¬ 
quires (unless the case is very exceptional) that a load shall be 
at one of the four points, A , B, C , D. 
To determine whether a given value of the stress is a maxi¬ 
mum or a minimum, and to determine which one among several 
maximum values will be greatest, without actually determining 
the values, requires a special discussion in each of the three 
cases above mentioned. It can usually be done, however, with 
little difficulty. 
MATHEMATICAL DISCUSSION. 
3. Proof of General Principle. —In deducing the general prin¬ 
ciple above stated, we shall at first refer to the case in which F 
falls between C and D (Fig. 2), so that n x , n 2 , l v l 2 are all posi¬ 
tive. We shall then consider the other cases, showing that the 
same formula applies to all. 
