Lamits of Economy of Material in Frame-structures. 589 
Final results.—The departure of the curves from a straight 
horizontal line shows the presence of overtones as well as 
the prime tone; while the fact that the curves appear to 
asymptote to this line shows that the overtones die away 
quicker than the fundamental. 
By both methods of experiment we see that the two curves 
agree best when p=°8 ; then substitute this value in (9) and 
we get a value for the frequency of vibration such that 
Ny ie 
= 986. 
I hope to make further experiments on tuning-forks or 
bars of acoustic frequency. I desire to render my best thanks 
to one of my colleagues, Dr. E. H. Barton, for his kind help 
in the work. 
University College, Nottingham, 
July 14, 1904. 
LVILL. The Limits of Economy of Material in Frame-structures. 
By A. G. M. Micuety, M.C.E., Melbourne*. 
AXWHLL has shown+ that for all frames under a 
given system of applied forces 
eet = Oe cates Al) 
where f, is the tension in any tie-bar of length J,, f, the 
thrust in any strut of length /,, and the first sum is taken for 
all the ties, the second for all the struts. C is shown to be 
a function of the applied forces and the coordinates of their 
points of application, and independent of the form of the 
frame. 
Starting from this result, we can find in certain cases lower 
limits to the quantity of material necessary to sustain given 
forces, and also assign the forms of frames which attain the 
limit of economy. 
If the greatest tensile stress allowable in the material 
which is to be employed is P, and the greatest compressive 
stress Q, the least volume of material in a given frame, 
consistent with security, is 
SPB tL, =. Me? is 
* Communicated by the Author. 
{~ Scientific Papers, 11. pp. 175-177. 
Phil. Mag. 8S. 6. Vol. 8. No. 47. Nov. 1904. Pe 
