384 MATHEMATICAL APPENDICES 17* 



u + Bu, v + Bv, w + Sw those velocities are to be understood 

 which replace the original values u, v, 10 at the same place (and 

 therefore for the same values of x, y, z], but at a different time. 

 From both systems of formulae it follows that to the equations of 

 condition, to which the variations Bu, Bv, Bw are subject, must be 

 further added the three equations 



= 2.m(y Bio z Bv) 

 = $.m(z Su x Biv) 

 = $.m(x Bv - y Bu). 



We take these formulae into account by multiplying them by 

 three provisionally undetermined but constant factors, which we 

 will denote by 2&, 2&iy, 2&, and adding them to the 

 former formulae, and then the coefficient with which each of the 

 3N variations Su, Bv, Bw appears to be multiplied in the sum of 

 the equations is to be put equal to 0. 



In this way, instead of the three differential equations that 

 stand at the end of 12*, we obtain the more general equations 



which may be integrated when u, v, w are taken to vary without 

 limit and x, y, z to be constant. 



This integration does not need to be carried out in order to 

 let us see that the result of the calculation will only differ from 

 that in the former case by the magnitudes 



zrj y, x zg, yg xrj 



having to be subtracted from the variables u, v, w as well as the 

 constants a, (3, y. The magnitudes have a similar meaning to 

 those of a, /3, y. While a, /3, y were found equal to the values of 

 the components a, 6, c of the velocity of the centroid, , rj, , as it 

 is easy to see, are the values of the angular velocities with which 

 the gaseous mass rotates about the axes of x, y, z ; for if this is 

 their interpretation, the sum 



a + zr 



