276 The Laws of Motion of a Cylinder, &c. 
a negative retarding force, is an accelerating force: 
acting in a contrary direction. Now the velo- 
city of the Block before the inftant of impact = 
Y4sx, when x == the diftance defcended, 
through by the Block in free fpace, and the velo- 
ow Tie 
city of the fyftem after impact == widens ; 
2s be the initial velocity, and gravity the retarding 
force, s will be the fpace defcribed: from this and 
what has been faid above this proportion will be 
a 45x? 
4 
eafily underftood, afqdec' hx qcanene REL 
a+qdceX ax a°x Cite 
A Xatqdce ~ hxatqdce me 
penetrated by the fyftem after the firft ftroke, which 
is therefore as x, that is, as the height defcended by 
the block, and may be put ==tx by writing ¢ for 
Sate Se See” tlie height defcended by 
hxa+qdce 
the block before the fecond ftroke == xx 1+t, 
Now in this procefs of penetrating the obftacle by 
repeated ftrokes, the quantity of matter, and the 
retarding force are given, confequently the fpaces 
penetrated, will be as the fquares of the initial 
velocities, that is, as the heights decended through 
by the block in free fpace ; therefore, asx:tx;: 
xXi-tt:txx<1+t = the fpace penetrated 
after 
