Motion of a System of Bodies. 47 



the same for the parallel plane as for that of %, y, as is evident by- 

 supposing the system to be reduced orthographically to the planes. 

 Now since the plane x, y is either at rest or moves parallel to itself, 

 its parallel plane will always be parallel to itself as the system moves: 

 it is also manifest that the first members of (22) will each =0, rela- 

 tive to any plane drawn through the same body (as before,) perpen- 

 dicular to the parallel plane ; Mec. Cel. Vol. I, p. 63. 



-Again, let, (for brevity,) any body of the system be indefinitely de- 

 noted by m, z, y, z, being its rectangular coordinates ; also suppose 

 (as in (d), (e), &c.) that z\ y\ z\ are the rectangular coordinates of 

 m, when referred to any other system of rectangular axes, which 

 have the same origin as the axes of z, y, z ; then denoting by a, 6, 

 &c. the same things as in (rf,) (e,) &c, we have by (d),z=az-\-by f 

 +cz', y=a'z'+b'y' + c'z\ z ~ a'V + b"y '+ cV, (a) : supposing 



dx 

 It 



the quantities in {a') to be functions of the time, we have 



z'da+y'db+z'dc adx'+bd y'+cdz' dy x'da' -f y'db' + z'dc' 



dt + dt '"^ = dt " + 



q ; (fo'+ b'dy'+c'dz' dz x'da" +y'db"+ z'dc" a"dx'+b"dy'+c "dz' 



dt ' dt~ dt + dt ' 



(&'). Put cdb+c'db'+c"db"=pdt, adc+a'dc'+ a"dc"= qdt, bda+ 

 b'da'+b"da"= rdt; then by (/) cdb + c'db'+c"db"= -bdc-b'dc' 

 ~b"dc"— pdt, adc + a' dc' + a" dc" = -cda-c'da'-c"da"= qdt, bda 

 +b'da'+ b"da"= - adb — a'db'-a"db"= rdt, (d') ; by substituting 

 the values of a, b, &c. from (o) in (d), we have sin. <p sin. 8d^ 

 cos. <pdd=pdt, cos. (p sin. ^ dT^-' -4- s i n . <pdd — qdt,di? — cos. 6d]<=rdt, (e'). 



dx' dy' dz' 



Put qz'-ry'+- d -=L, rz'-pz>+ -£ =M, py' - qx' + £ =N, (/') ; 



then multiply (b') by a, a', a", respectively, add the products, and 



u u f f\ fjrt t'm adx + a d y '+ a " dz t • VI 

 we have by {j ), [a'), (j ), -, = L, in like manner 



bdz+b'dy+b"dz cdx+c'dy+c"dz „ . 



J t =M, dl =N, (g') ; multiply (g') by 



dx 

 a, b, c severally, add the products, and we have -7:=aL-{-&M-f-cN, 



dz < 



and m like manner -f t =a'L+b'M+c'N, -j t = a"L -{-b"M + c" N , (h'). 



By adding the squares of (A'), we have by (/) ~- ^ ~ — L * 



+M»+N* ; hence by (/'), ^±|l±^« Cf 4 *^*^ 



