424 



Mb. O'BRIEN, ON A NEW NOTATION FOR VARIOUS EQUATIONS 



other particles, must satisfy the conditions of equilibrium. We have therefore by equations (28) 

 and (29), 



Let u be the symbol of the centre of gravity of the body, and assume ti = u + u ; then these 

 equations become (observing that 'S.uSm = 0), 



dt- 



df 



Which equations are equivalent to the six equations of motion of a rigid body. 



Since u is the symbol of Sm with respect to the centre of gravity as origin, the second of these 

 equations determines the motion of rotation of the rigid body about its centre of gravity, and, as 

 far as this equation is concerned, the centre of gravity may be regarded as a fixed point. 



d „ du ^du du ^ d'u d'u 



Also, since — D u . -— = D -~ . , + Du . -^, = Ott . — — , 

 ^ dt dt dt dt dt- dt' 



this equation may be written in the following form, 



— fs/>«.^^ml = Si?«.C/^>« (.-ifi). 



dt \ at ) 



27. To effect the integration denoted by 2 in the first member of equation (36). 



Take the principal axes through the centre of gravity as the co-ordinate axes, and let j-, y, x, 

 be the co-ordinates of Im : then we have 



u = xa + yfi + ^7, 

 and therefore, since .r, y, x are independent of t. 



du 



.(37). 



da dii rf-y 



= X — + V — + z —^ 

 dt dt ^ dt dt 



Now, referring to Art. 2, we may see immediately, that, if u), denote the velocity of the 

 point B parallel to OC, u>: the velocity of C parallel to OA, and w^ the velocity of A parallel to 

 OB (in other words, oi,, ui-i, 0)3, are the angular velocities about the axes OA, OB, OC, of the 

 planes BOC, COA, AOB respectively), then we have 



a = u>,dt, b = wodt, c = w:,dt, 

 and therefore the equations (3) become 



— =a.3pJ -«.«7* 

 o t 



-—- = a),7 - (U3C 

 at 



d^ 

 d 



dy 



dt 



= coja — <0| 



/3 



.(38). 



We may here observe in passing, that, if we assume 



■(3yj, 



• If we put these values in (3?) the coefficients of a, p, y are •«.iz-u,,y, ,.,,.v-m,x, „., j/-,.,a,r ; wliich are the well kiu.un exprev. 

 sions of the velocities of any point of a rif,'id body moving about a fixed point. 



