Mathematical Discussion. 
81 
comes applicable if in Figs. 1 and 3 the directions of the loads 
and reactions is reversed; the effect being merely to reverse the 
stress in every member. 
If in Figs. 1 and 3 the member H L had been considered instead 
of H K, the resulting expressions would be of the same form as 
those already given; but it is to be noted that when H K is in 
tension, H L is in compression, and vice versa. 
General Result 
We have now shown that equation (7) is a general ex¬ 
pression for the condition for maximum values of the stress in 
any member of a truss of the kind discussed, under any distri¬ 
bution of loads which does not make w infinite at any point. 
We shall next consider the general condition for discriminating 
between positions giving maximum values and those giving 
minimum values of the stress; we shall then discuss the case of 
concentrated loads; and finally w T e shall consider the problem of 
determining which one, of several positions in which equation 
(7) is satisfied, corresponds to the true greatest value of the 
stress. 
4. Discrimination between Maxima and Minima. — From equa¬ 
tion (6) we have, by differentiating, 
= l 2 a' + l 1 b' — — ( n 2 c' + n L d') 
If this is negative, the value of T is a maximum; if positive, a 
minimum. 
The application of this condition in special cases will be con¬ 
sidered below. 
5. Concentrated Loads. — By a concentrated load, in the strict 
mathematical sense, is meant a finite load applied at a geomet¬ 
rical point. At the point of application of such a load the 
value of to (the load per unit length of the truss) becomes infi¬ 
nite, and the above discussion in its present form is not appli¬ 
cable. As regards train loads actually applied to bridges, 
however, it is to be remarked that these are not concentrated in 
