DE. PLUCKEE OX FUNDAMENTAL VIEWS EEGAEDING IVIECHANICS. 373 



for the rotation being (tuO, tuv), its coordinates are 



0, 0, V, uv, 0, 0. 

 Accordingly OZ is the axis of the moment ; we obtain 



and in putting " 



^v= — tana;, 

 we have 



tano) tanw. 



4> — 8~' yi= ^j2- ' 



Here u denotes the a7igle of rotation, taken in starting from the plane acted upon in 

 the direction from OZ to OX, In passing to infinitesimals, the last equation becomes 



7. When two rotations take place simultaneously, there is a resulting one in the case 

 only where both axes of rotation are confined within the same plane. Let 



X, 8), 3, m-3u, 3t-Xv, 3EW-P, 



X', 8)', 3', ^'v'-3'u', 3't'-rv', $V-g)V, 



be the six coordinates of the rotation, (t, u, v) and (f, u', v') being any two planes containing 

 their axes. If both axes be confined in the same plane, t', u\ v' may be replaced by 

 t, u, v. In this supposition, by adding the corresponding coordinates, we get 



X+s:', 8)+8)', 3+3', 

 (g)+8)')"-(3+3>, (3+3')^-(3£+2e>, (2e+3£'>-o+8)% 



These six sums are the coordinates of a new rotation, the axis of which is within the 

 same plane (t, u, v). Here the three equations of condition, 



x+3£'=o, g)+8)'=o, 3+3'=o,: 



which render the six coordinates of the resulting rotation equal to zero, are sufficient to 

 express that equilibrium exists. 



In the general case, where both axes of rotation are not confined within the same 

 plane, the six sums of coordinates 



Si+x', §)+§)', 3+3', 

 (g)?;+g)V)-(3?t+3'«'), (3^+3'<')-(x«+3eV), (3cu+x'u')-(^jt+^'q, 



are the coordinates of a dyname. When equilibrium exists we get, in order to express 

 that all resulting effect be destroyed, six equations of condition by putting the six 

 coordinates equal to zero. 



8. By generalizing, the following theorem is immediately derived : — 



Any numher of rotatory forces acting simultaneously, the coordinates of the resulting 

 rotatory force, if there is such a force, if there is not, the coordinates of the resulting 



