£78 Charges of greatest Efficacy for Artillery at Sea , 
Lemma L 
If two spheres of different diameters and different spe- 
cific gravities impinge perpendicularly on two uniform 
resisting fixed obstacles, and penetrate into them ; the 
forces which retard the progress oj the spheres will be as 
the absolute resisting forces or strengths of the fibres of 
the substances directly, and the diameters and specific gra- 
vities of the spheres inversely . 
Let B and r denote the absolute resisting forces of the 
two substances ; F and / the retardive forces; B, d the 
diameters of the spheres ; Q, q their quantities of matter, 
and N, n their respective specific quantities. Then the 
whole resistances to the spheres, being by mechanics 
proportional to the quantities of motion destroyed in a 
given time, will be as the absolute resisting forces of the 
substances and quantities of resisting surfaces jointly ; or 
as the resisting forces of the substances and squares 
of the diameters of the impinging spheres : that is, 
M B, W 
m r X d 2m 
M F Q 
But in general — j-x — 
F 
two values of the whole resisting forces, we obtain 
therefore equating these 
Q _ Jt B 
d 3 
d 2 
n 
F B B 2 
and jr=— x -^x 
U D 2 d 3 
X -fX-tu x 
/ 
and since -ft— 
Q 
R 
r 
x 
n 
£ _ 
D X N 
± 
*1 
F___B jD 2 £ n_ 
| F A N, 11 13 / r x d 2 x B 3 x N : 
that is, the forces retarding spheres penetrating uniform 
resisting substances are as the absolute strengths of the 
fibres of the substances directly, and the diameters and 
specific gravities of the spheres inversely. 
