Economy of Material in Frame-structures. D97 
equal and directed to a point on the line of action of the 
third outside the triangle formed by the points of application 
of the forces. Fig. 4 is an example. 7 
The sectors DUG’, FCE’ undergo uniform bulk compres- 
sion, the remainder of the field pure shear as in fig. 2. 
The volume of the frame is 
Fx { CExax (5 +5) +(AE+BG)x 6 b 
where a is the angle ECG. 
4. (Fig. 5.) Equal and opposite couples applied at points 
A, B on the straight line AB. 
The minimum frame consists of the series of rhumb-lines 
inclined at 45° to the meridians of the sphere having its 
poles at A and B. 
All the rhumb-lines of one series and all similiar lines on 
spheres concentric with the spherical frame are uniformly 
and equally extended, all the lines orthogonal to these on the 
spheres equally compressed. 
The third system of orthogonals, viz., radii from the centre 
of the sphere, are unchanged in length and simply rotated 
about the axis AB. 
The minimum volume of the frame is 
2, fan *) 1 5) 
Pangan (F +3) (+5) 
where L is the moment of the transmitted couple, 20=AB, 
and »X is the latitude of small circles about each pole 
corresponding to the circle of radius 7) in Case 1. 
Melbourne, April 19, 1904. 
