040 
MECHANICS, 
Now, by fublHtuting in the place of v its value as deter¬ 
mined in Cafe 2. Prop II. we obtain 
: V- 
=V 7 - 
1 B'V— B'V 7 
B + B 7 * 
1 -j-«X BV—BV 7 
i B-UB 7 
Cor . t. Hence, by converting the preceding equation 
into analogies, B-J-B' : l-j-^xB as the relative velocity of 
the bodies before impart is to the velocity gained by B' in 
the direction of B’s motion ; and B-J-B' : i-j-»X B 7 as the 
relative velocity of the bodies before impact is to the velo¬ 
city loti by B. 
Cor. 2. The relative velocity before impart is to the re¬ 
lative velocity after impart as the force of compreffion is to 
the force of reftkution, or as 1 : n. The relative velocity 
after impact is v "■—; or, taking the pre ceding value of 
, , 1 4- n X B V—B V'l 
quantities, —*■' — XTI ' * 1 
thefe 
: V'. 
B-f B' 
1 + n X B'V—B'V' 
iV'—V 
, I+B x B +B'x v—V' 
B + B' 1 B+B' ’ 
dividing by B-{-B', we have v" — v'~V —V+V—V' + 
«XV—V 7 22«XV—V'— the relative velocity after impact. 
But the relative velocity before impact is V — V 7 , and 
V—V' : n X V—V'=ri : n. The quantity V 7 has evidently 
the negative fign when the bodies move in oppolite di¬ 
rections. 
Cor. 3. Hence from the velocities before and after im¬ 
part we may determine the force of reftitution or elafticity. 
Prop. XXVIII. When a perfeElly-hard body impinges ob¬ 
liquely on a perfeElly-hard and immoveable plane A B, in the di¬ 
reElion C D, after impaEl it will move along the plane , and, The 
velocity before impaEl : the velocity after :: radius : the cofine 
of the angle C D A.—Take C D (Plate IV. fig. 53.) to re- 
prefent the motion of the body before impart ; draw C E 
parallel, and D E perpendicular, to A B. Then C D may 
be refolved into the two, C E, E D, of which E D is wholly 
employed in carrying the body in a direction perpendi¬ 
cular to the plane; and, fince the plane is immoveable, 
this motion will be wholly deftroyed. The other motion 
CE, which is einployed in carrying the body parallel to 
the plane, will not be afferted by the impart ; and confe¬ 
quently, there being no force to feparate the body and 
the plane, the body will move along the plane ; and it 
•will defcribe D B =C E in the fame time that it defcribed 
CD before impart; alfo, thefe fpaces are uniformly de¬ 
fcribed ; confequently, The velocity before impart : the 
velocity after :: CD : CE :: radius : fin. ^ CDE :: 
rad. : cof. f C D A. 
Cor. The velocity before impart : the difference be¬ 
tween the velocity before and the velocity after, that is, 
the velocity loft :: radius : rad— cof. Z. CD A :: rad. 
; the verfed fide of the angle C D A. 
Prop. XXIX. If a p erf Elly-elafiic body impinge upon a 
perfeElly-hard and immoveable plane AB, in the direElion C D, 
it will be refeEled from it in the direElion D F, which makes, 
with D B, the angle B DF equal to the angle A DC.—Let 
CD (fig. 54..) reprefent the motion of the impinging 
body; draw CF parallel, and DE perpendicular to A B; 
make E F — C E, and join D F. Then the whole motion 
may be relolved into the two C E, E D ; of which C E is 
employed in carrying the body parallel to the plane, and 
mull therefore remain after the impart; and ED carries 
the body in the direction ED, perpendicular to the plane ; 
and, fince the plane is immoveable, this .notion will be 
deftroyed during the compreffion, and an equal motion 
will be generated in the oppofite direction by the force 
of elalticity. Hence it appears, that the body at the point 
D has two motions, one of which would carry it uniformly 
from D to E, and the other from E to F, in the fame 
time, viz. in the time in which it defcribed CD before 
.the impart j it will, therefore, defcribe P F in that time. 
Alfo, in the triangles C D E, E D F, C E is equal to EF, 
the fide E D is common, and the / CED is equal to the 
/_ D E F ; therefore, the /CDE = the / EDF; hence, 
the / CDA = the /FDB. 
Cor. 1. Since CD—.DF, and thefe are fpaces uniformly 
defcribed in equal times, before and after the impart, the 
velocity of the body after reflertion is equal to its velocity 
before incidence. 
Cor. 2. If the body and plane be imperfertly elafiic, 
take (fig. 55.) DE : Dt :: the force of compreffion : 
the force of elafticity; draw xf parallel and equal to 
E F, join F/, D/; then the two motions which the body 
has at D are reprefented by D x, x f, and the body will 
defcribe D f, after reflertion, in the fame time that it de¬ 
fcribed C D before incidence; therefore,The velocity be¬ 
fore, incidence : the velocity after reflertion :: C D : D/ 
:: DF : Df :: fin. D/F, or fin. of its fupplement E Df 
: fin. D F /, or fin. F D E :: fin. ED/: fin. E D C. 
Prop. XXX. To fnd the point of an immoveable plane 
which an elafic body moving from a given place muf frike, in 
order that it may, after ref exion, either from one or two planes, 
impinge againf another body whofe pofition is given. 
1. When there is only one refexion .—Let C (fig. 56.) be 
the place from which the impinging body is to move, 
and let E be the body which is to be Itruck after reflexion 
from the plane A B. From C let fall C H perpendicular 
to A B ; continue it toward G till HG=CH, and join 
G E by the line G DE; the point D were this line cuts 
the plane, is the place againlt which the body at C mull 
impinge, in order that, after reflexion, it may ftrike'the 
body at E. The triangles C D H, H D G, are equi-an- 
gnlar, becaufe two fides and one angle of each are refpec- 
tively equal; therefore the angles DCI-I, D G H, are 
equal. But, on account of the parallels FD, C G, the 
angle EDF=DGC = DCH, and DCH = FDC; 
therefore the angle of incidence FDC=FDE the angle 
of reflexion; confequently by Prop. XXVII. a body 
moving from C, and impinging on the plane at D, will, 
after reflexion, move in the line DE, and ftrike the body 
at E. 
2. When there are two ref exions .—Let A B, B L, (fig. 57.) 
be the two immoveable planes, C the place from which 
the impinging body is to move, and F the body which it 
is to ftrike after reflexion from the two planes; it is re¬ 
quired to find the point of impart D. Draw C H G per¬ 
pendicular to A B, fo that H G = C H. Through G 
draw GMN parallel to A B, cutting L B produced in M, 
and make G M — M N. Join NF; and from the point 
E, where N F cuts the plane B L, draw E G, joining the 
points-E G ; the point D will be the point of the plane 
againft which the body at C muft impinge in order to 
ftrike the body at F. By reafoning as in the preceding 
cafe, it may be fhown that the angle CDH = EDB, 
therefore D E will be the path of the body after the firft 
reflexion. Now', the triangles G E M, E M N, are eouian- 
gular, becaufe G M =2 M N, and the angles at M right, 
therefore DE B 22 F E L, that is, the body after reflexion 
at E will ftrike the body placed at F. 
Prop. XXXI. Having given the radii of two fpkerical 
bodies moving in the fame plane , their velocities, and the di- 
reElions in which they move, to fnd the plane which touches 
them both at the point of impaEl. — Let A E, BE, (fig. 58.J 
meeting in E, be the directions in which the bodies A 
and B move ; and let AE and B D be fpaces uniformly 
defcribed by them in the fame time ; complete the paral¬ 
lelogram A B K E ; join K D, and with the centre E and 
radius equal ro the fum of the radii of the two bodies, de- 
fcribe a circular arc cutting KD in H; join E H, and 
complete the parallelogram EH MR. Then R av.d M 
will be the places of the centres of the two fpheres when 
they meet; and, if RC be taken equal to the radius of the 
fphere A, the plane C L, which is drawn through C per¬ 
pendicular to M R, will be the plane required. Since 
M H is parallel to AE or B K, the triangles D M H, 
D B K, are funilar, and B K : B D ;: M H : M D ; or 
AE 
