594 Mr. A. G. M. Michell on the Limits of 
lines of principal strain on which material bars are not required 
are shown by dotted lines. 
1. (Fig. 1.) A single force F applied at A, and acting at 
right angles to the line AB, is balanced by an equal and 
opposite force and a couple, of moment F x AB, applied at B. 
Fig. 1. 
The minimum frame is formed of two similar equiangular 
spirals having their origin at B and intersecting orthogonally 
at A, together with all the other spirals orthogonal to these 
and enclosed between them. 
All the spirals of one series are equally extended, those of 
the other system equally compressed to the same degree. 
The spirals may be extended to infinity, and the infinite field 
will remain continuous with equal and opposite strains in the 
principal directions at all points. The necessary volume of 
material is infinite if the frame is continued to a mere point 
at B; but if forces equivalent to the force I, and the couple 
Fx AB at B, are distributed over a small circle with its 
centre at B, and of radius 79, the volume necessary is 
ae Ee 
Fa log <. (5 i a), 
where a=AB, and P, Q are the stresses allowed for tension 
and compression. 
2. Centrally-loaded Beam.—(Fig. 2.) A single force, F, is 
applied at C, the middle point of the line AB, and is balanced 
by equal parallel forces at A and B. 
