Johnston — Initial Motion. 17 



co-ordinates of every particle m of the system can be expressed as a 

 function of them and the time. 



We have, then, if X, Y, Zhe the components of the applied forces 

 acting at any point, and Xi, Yi, Zi, of the initial reactions due to the 



constraints. 



2mSx = 2 (X + Zi) 5;, 



2wSy = 2(r+ ri)s^ 

 2w5i =2(Z + Zi)St. 



Therefore, or by the principal of virtual velocities for any small 

 displacement Sz, Sy, Sz, we have 



^m{U5x + 5y5i/ + Sz5s) = ■S{(X + Xi)Sx+ {¥+ Yi) Si/ + {Z + Zi) dz} St. 



Again, since 



dec dx 



— - = -T, &c., 



d^ dO 



dx ^ dy . dz\ I _ dx . dy . dA\ 



I dx dy dz\ I 



Si: -7 + Sy -^ + 5z — 

 dQ de de I 



I dx dy dz \ 



\ de ^ de del 



(,^dx .^ dy .^ dz\ 

 xS — + yS -^ + zS — ) 

 de de de I 



(dx dy .dz\ 

 X—: + y^ + 3 — r . 

 de de del 



For X, y, z are each zero initially, and, consequently, iS —r, &c., 



SO 



are, at most, small quantities of the second order, and may be 



neglected. 



"We have, then, 



/ dx dy dz\ dT 



2w 5a; — + 5y — + 5i — =5 — , 



\ de de del de 

 where T is, as usual, the kinetic energy of the system. Therefore, 



dT dT „ dT 



2w {SxSx + 5y 5y + 5i5z) = 5 — r- 59i + 5 -^ 5^2 + &c. + 5 — d(pi + &c. 



dei de2 d(pi 



Let ©iS^i, ©28^2; ®n+iS<^i. &c., be the virtual moments of the applied 

 forces found by varying 61, O2 . . . (fix, ffi2 • ■ • &c., separately (vide 

 Routh, "Elementary Eigid Dynamics," § 426). Also let 4>i, $2 . . . 

 ^m be the initial reactions due to the smooth constraints. They give 



E.I.A. JilOC, SEE. III., VOL. in. C 



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J 



