590 Mr. A. G. M. Michell on the Limits of 
Now, considering a number of different frames in equi- 
librium under the same set of external forces, in that one of 
the frames in which V is least 
2PQ.V+(P—Q)C is also least; 
ie, POLS. +E. |} +(P-QIS. LL - 3 lhl 
=(P+Q){>.47,4+2.1,7,} is least, 
or >. ULF] is least, ¢)S0pe) kc, Sr 
where [f] denotes the numerical value of the force in any 
bar, and the sum is taken for both struts and ties. 
One proof given by Maxwell of the equation (1) above 
depends on consideration of the virtual work of the applied 
forces and internal stresses during an imposed uniform dila- 
tation or contraction of the frame. Consideration of a more 
general type of imposed deformation will furnish information 
as to the quantity %./ [/}. 
Let the space within a given boundary, which encloses a 
number of different frames subjected in turn to the same set 
of applied forces, undergo an arbitrary deformation of which 
the frames partake and such that no linear element in the 
space suffers an extension or contraction numerically greater: 
than e.6/, where d/ is the length of the element and ¢€ a given 
small fraction. 
In this deformation, let any bar of length / of one of the 
frames, A, undergo the small change of length e/, which is 
to be taken as positive when it increases the existing strain 
in the bar due to the applied forces, negative in the contrary 
case. The increase in the elastic energy of the bar is e/f,. 
and for the whole frame 
>. elf=sW, 
by the principle of Virtual Work, where 6W is the virtual 
work of the applied forces, and is independent of the form of 
the frame A. 
Thus SW= 2 .elfp>. [ell] 
$S.cl[/] ped. 1[/] 
OW 
PS Fe eae 
indicating by the suffixes that the inequality applies to the. 
frame A. 
If, however, a frame, M, can be found, such that all its 
parts have their strains increased equally and by as much 
or 
