246 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. 



coincides with that of the moving axes at some particular instant 

 of time, the direction cosines vanish with the exception of a 1} fl. 2 , y s , 

 which are equal to unity. We then have 



-dt d dt --dt- dt 



But since /3 3 = cos (?/'), y 2 = cos (#?/), we have on differentiating 



d&* f i\ d(iiz') dy . f , N d(zy') 



-/f - - sm (^ ) T^fr' w = ~?* ('y ) r&-> 



and since 



sin(^) = sin(^)=l, f = 



Thus it is clearly seen that p, q, r are angular velocities, being the 

 rates of increase of the angles #/', x#', yx', or in other words, the 

 angular velocities with which the moving axes X, Y, Z are turning 

 about each other. 



It is to be noticed that p y q, r, though angular velocities, are 

 not time -derivatives of any functions of the coordinates, which might 

 be taken for three generalized Lagrangian coordinates q. 



They are merely linear functions of the derivatives of the nine 

 cosines, which latter may themselves be expressed in terms of three g's. 



If we seek to find those points of the body whose actual velocity 

 is a minimum, we must differentiate the quantity, 



121) v * = (V x + q z- r yy + (V y + rx -p^ + (V z + py-qxy 



with respect to x 9 y, z, and equate the derivatives to zero. We 

 thus obtain 



r(V y + rx pz) q(V z +py qx) = 0, 



122) p(V z +py - qx) -r(V x + qz- ry} = 0, 

 q ( V x + qz r y) p ( V y + rx - pg) = 0, 



which are equivalent to the two independent equations, 



p q r 



These are the equations of a line in the body, namely of the central 

 axis, as found in 66, 38). 



Calling the value of the common ratio A, clearing of fractions, 

 multiplying by p, q, r, and adding, we obtain the value of ^,, 



n Y _L T/ _i_ *. 



124) i-*S 



