Economy of Material in Frame-structures. D938 
the change of direction of R, between 8, and Sz, and let X/ 
be the extension of the R curves. 
Then, as before, &r=N/r; and making a circuit around 
the element, since the angles remain right angles, 
db a, ON 
Sa me ar b=: 
but —.n—-—* .1l=0, because the angles were right angles 
before the deformation. Thus since \/X, by assumption 
d d 
a = and ov =0. 7) QHD: 
There are two general classes of orthogonal curves satis- 
fying the required conditions, viz. :— 
I. Systems of tangents and involutes derived from any 
evolute curves. 
(Since such systems are bounded by the evolute curves, 
the corresponding frames are in general of minimum volume 
only relatively to others with the same finite boundaries.) 
II. Orthogonal systems of equiangular spirals, with systems 
of concentric circles and their radii, and rectangular 
networks of straight lines, as special cases. 
Frames whose bars coincide with the curves of any of 
these systems are therefore frames of minimum quantity 
of material, for any system of forces consistent with their 
equilibrium and continuity of displacement. 
Frames may also be built up of parts of different systems 
of these classes provided that continuity of displacement is 
secured along the lines of junction. 
Hxamples of such frames of minimum quantity of material, 
for some elementary systems of forces, are given in the 
accompanying figures. 
Compression bars in each case are indicated by thick, and 
tension bars by fine, continuous lines. Those portions of the 
