43 



where the vector-symbol t denotes a straight line drawn in 

 the direction of the axis of momentary rotation, and having a 

 length which represents the angular velocity of the system ; 

 80 that this vector t is generally a function of the time t, but 

 is always, at any one instant, the same for all the points of the 

 body, or of the rigid system here considered. The equation 

 (12) thus gives, by an immediate integration, the following 

 expression for the law of areas : 



S.WaF. <a = 7 + S.»wrja^d<; (14) 



where -y is a constant vector ; and if we operate on the same 

 equation (12) by the characteristic 2S\i(\t, we obtain an ex- 

 pression for the law of living forces, under the form : 



S.m(r. m)2 = -AV2S./w5jta^d<; (15) 



where A is a constant scalar. The integrals with respect to 

 the time may be conceived to begin Mjith t = ; and then the 

 vector y will represent the axis of the primitive couple, or of 

 the couple resulting from all the moving forces due to the 

 initial velocities of the various points of the body ; and the 

 scalar h will represent the square root of the primitive living 

 force of the system, or the square root of the sum of all the 

 living forces obtained by multiplying each mass into the square 

 of its own initial velocity. Again, the equation (13) gives, by 

 diflPerentiation, 



d^a j^ da ^^ di dt 



d?^=^"d7^^-d7" = '^-'«-^-"d-.' (^^^ 



and for any two vectors a and t, we have, by the general rules 

 of this Calculus, the transformations, 



V.a{iV.ta)-V.i{aV.La) = \V.{Lay\ 



= S .la .V.ia = \V. i{aia)=-\V.a{iai);) ^ '' 



therefore, by (12) and (14), 



r dl „ 



J 



2 . ma V . a J- + ^ .m r. a<p = F. il. . ma F. la 



(18) 



