and Strength of Materials 231 
(LC. (42) teats 
Whiere if v =0, as in Galileo’s hypothesis, 
a=h, orthe whole beam is stronger than 
any partofit. If againv=1, as with Ma- 
riotte, we have a= .76759.h where nearly 2th 
of the height may be cut off from the top, 
and the remainder be as strong as the whole 
beam. 
9. If however we should wish to find the 
strength of a beam whose form was such that 
the forces exerted by its fibres could not be 
sh ha? a3 
* ss is thus:—Si —— (——~ —~—— J= 
The process is thus :—Since : (as = 
“(sae SE aE , divisi i- 
F] ‘ie ae a we have by division and transposi 
3 
tion a3 — 2 hat + (= ~ 1) h3 = o, a cubic 
equation : But as onc of its roots is evidently = h, we 
may, by dividing it by a—A, depress it to a quadratic ; 
sim . ha? +(5- 1 )A3 
Pee Se (putting n for 
eae nha* + (n— 1) 43 
aa)= - =a? — (n— 1) ha — 
(n—1) 4? =0. 
a-—h 
And sine ahe for n, we have 
as (+ qe (73)" +53|+ rey) h, as above. 
