48 Motion of a System of Bodies. 



+2' a )? 3 + (z' 2 +y' 2 )r*-2y'z'.qr-2x'z'.pr-2x'y'.pq+2\y- -jt 



,z'dx'-xdz'\ ( x' dy' — y'dx' \ dx'*+dy' 2 + dz 2 ,. x 



[xdy — ydx\ 



Now (18) are easily changed to d.Sml jr — ) =di. Sm(Qx 



zdx — xdz 



Py), d.Smi j— J = dt. Sm(Pz-Rz), d.Sm[~—^-—\ = dt. 



z—z 



dy 



Sj»(Rv - Q 



A, S(ar' 2 + ^ 2 )«* = B, 



S(x'*+y' 2 )m=C, Sy'z'm^D, Sx'z'm^E, Sx'y'm=F, (k% also put 

 Ap - Er - Fgr = p', Bq-Dr -Fp = q', Cr ~Vq- Ep= r', 



f x'dy' - y dx'\ ( z'dx'-x dz\ tydz'-z'dy 

 Sm ( If— j = < A ' Sm [ dt ) =' B ' Sm [ dl 



,C, (l). By substituting the values of x and y from (a) and those 



dx dy 

 of -j. » j , from (fr), in the first of (23), we shall have (since a, 6, 



&ic., are the same for all the bodies;) d. [{be'— c6').Sm(Ny' — Mz') 

 4-(ac-ac) . S/n(Lz'- Nx') + (a6' - 6a) . Swi(Mx'-Ly')] = d* . 

 Sm(Q:r-Py,) (24). 



PmdLSm(Qx-Py)=dN' ", dr.Sro(P*-frr)=:rfN", Jr.Sm(% 

 Q^)=JN', (»i') ; by substituting the values of L, M, N, from (/'), 

 we have Sm{N y - Mz' ) = p' + £ , Sm(L^'-N.r')=^+ / B, Sm(Mx' 

 — Ly)=r '+,A, (n'), by substituting these values, and those of 6c' 

 cJ',&c, from (#), and using dN'", (24) becomes «\[>"(y-f ,0) + 

 6"( 9 '+,B)+c"(r'+,A) ]=dN'", in like manner the second and third 

 of (23) will give, d.[a'{p'+ ,C) + 6(?' + ,B) + c '( r '+ l A)]=dN", 

 <*. [«(/ + £)+ 6(2'+ / B)+c(r'+ ,A)] = rfN', (25) ; the two last of 



these are easily derived from the first by making some very obvious 

 changes in (24). 



By taking the differentials indicated in (25), then multiplying the 

 resulting equations by a", a', a respectively, and adding the products, 



we shall have by (/) and (d'), after dividing by dt, , / — -f 



*i , , x>x «''<ZN"'+«'dN''+adN' . • 

 ^(r'-f /A) - r(5' / + / B)= ^ , and in a similar way 



^— + r( P ' + ,C) - p{r> + ,A) = ^ 



«\(r'+,A) , , ■ . . ^ c'W'+cW+cdN' 

 - L -rfT-+K3 / +-B)- 9 ( F '+,C)= -^ , (26). 



Put the right hand members of (26) equal to N„ N„, N„, respec- 



