292 CELESTIAL MECHANICS. I. vii. 33. 



For since --—u,-^=.v, and — ^ w, if we take the 

 dt dt dt 



fluxions of these equations, regarding ?/, v, and w as func- 

 tions of the coordinates x, y, and z of the particle, and of 

 the time t, we shall have 

 ddjr d'u , d'u . d'M d'M 

 dt^-^dt d^ dy ^^"^dz ' 



ddy d'v , d'v , d'v , d'v , 



dF^=dF + "S + "d^+"'cb = ""'^ 



dd2 d'w , d'l/; , d'u , d'w ^^ 

 -5-^-=-^-+ «^-T— + ^-r- +^-r-- [For since 

 Qt^ dt djp dy dz 



du=: __d# + -— dxH dij + — dz, and dxzzudt, dyzzvdt, 



dt dx dy dz 



and dzzzwdt, the truth of the equations is manifest; and 



by substituting these values in the equation (j^) 363, we 



obtain the equation {H) of this proposition. ] 



367. Theorem. For the equation of con- 

 tinuity we have also o=i^+^V€^V^"->. 



•^ d^ dx dy dz 



(K) 



If we suppose the coordinates, x, y, and z, to be infi- 

 nitely near to a, &, and c, we may conceive a, b, and c in 

 the value of €*, to be equal to x, y, and z, and x, y, and z 

 to become x + uAt, y-\-vAt, andz-\-ivAt: we shall then have 



Czil + aM -—+-—+——); [since -— becomes-— + 

 Vda: dy dz / da da 



d'ti d'x . d'l/ ^ , ^ dM dV 1 . ^ d'u , d'z 



da ^^ da: dx dx db dy dc 



I 



