728 PHILOSOPHICAL TRANSACTIONS. [aNNO 1780. 



A X AD^ + B X BD" , B X BD X AB , 



also CA = CD + DA = _---— ^^-^ + DA = -_--—-—_ ; heiicc 



B X BO X AB ._ EO ^^^^^ ^^ BXBDXGB+AXADXA O __ 



B X BD — A X AD ^ ' A ^ ' B X BD — A X AD 



B X BD X c.B + A X AD X AG ^ ^^^ ^j^^ velocitv of tiie centre of gravity : hence if 



AXBXAB ■' a J 



B y G B"' -^ \. V A f "^ 



the motion be communicated at g, the velocity becomes as ^ . 



•' A X B X AB 



Let now the motion, which is supposed to be actually communicated to the rod 

 at D, be equivalent to the motion of a body whose magnitude is g, and moving 

 with a velocity v ; then if that motion be communicated at g, the velocity of the 



r. -^ ■ 11 1 i 1 G X t' 1 B X BG' + a X AG^ 



centre oi gravity is well known to be = ; hence : 



O J A + V AXBXAB 



B X BD X EG + A X AD X AG _ G X V ^ G X V B X EG X B D + A X A D X AG 



AXBXAB ''a+b'a + B BXKG-+AXAG'^ 



the velocity of the centre of gravity, when the same motion is actually com- 

 municated to any point d. Now bd = bg + gd, and ad = ag — gd ; hence 



B X BG X BD + A X AD X AG = B X BG' + A X AG^ + GD X (b X BG — A 



X ag) = (because b X bg — a X ag = o) b X bg" + a X ag' ; consequently 



the velocity becomes ; and hence the centre of gravity moves with the 



same velocity, wherever the motion is communicated. 



Prop. 3. Let a given elastic body p, moving with a given velocity, be supposed 

 to strike the lever at the point d, in a direction perpendicular to it; to determine 

 the velocity of the centre of gravity g after the stroke. — Suppose first the body to 

 be non-elastic, and let v be the velocity of the centre of gravity after the stroke 

 on that supposition, cid v the velocity of the striking body : then cg : cd : : ti : 

 "LiLEf = the velocity of the point d after the stroke, or of the body p : for the 



CG 



same reason - -— and equal the velocities ot a and b respectively. 



Now because, in revolving bodies, the momenta, arising from the magnitude of 

 the bodies, their distance from the centre of rotation and velocity conjointly, 

 remain the same after the stroke as before, we shall have p x v X dc = 



V X CD^ X P , V X ex'- X A , V X CB'- X B i .1 r 



—- , and therefore v = 



CG CG CO 



PXVXDCXCG PXVXCC 



„ — , , N , -• Hence if p be sut^posed 



P X DC^ + A X AC^ + B X r,C* (a + B) X CO + 1' X DC ' t^ 



O s/ p s/ V y PC 



an elastic body, we shall have _^ ^^ x cg + p x do ^°^ ^'^^ velocity of the 

 centre of gravity after the stroke, in ipso motus initio. 



Prop. 4. To determine the motion of the bodies after the first instant, or when 

 they are left (o move freely by themselves. — The writers on mechanics, from 

 considering the equality of motion on each side of the centre of gravity, when a 

 body revolves about that point, have inferred, that if a body had a projectile as 

 well as a circular motion communicated to it, the centre of gravity \vould con- 



