Joiii!iSTOi!f— Initial Motion. 15 



If «', b', c', /', ff', h' are the first minors of the determinant, 



a — h — g I 

 A= -A i -f \ 

 -9 -f c I 

 we have, from equations (11), 



ASwa: = (-P - xh'Bz + a;^'i?2) Sif, 

 ASwj, = (Q - ici'-Bs + xfEz) 5t, 



A5«z = (5 - ir/'J?3 + xc'Ii2) 5t ; 

 where 



Ps a'Z + h'M + g'N, &c. &c, 



"We are at liberty to take as our axes of y, z any two lines at right 

 angles to each other in the plane, through the centre of inertia, per- 

 pendicular to the normal to the rough surface. "We will take our new 

 axes subject to this condition, and so that the new value of /' = 0. This 

 can always be done ; for if the new axes make an angle 6 with the old, 

 the new value of /' is 



/' cos 26 + {c' - b') sin d cos 0. 



Therefore the new axes of y and 2 are the principal axes of the 

 section of the ellipsoid of gyration made by the plane perpendicular to 

 the axis of x. 



Now, supposing all our equations referred to these axes, we have 



ASuy = (Q - xb'Hs) 5t, 

 ASwz = (i? + xc'£2) S(, 



Substituting these values for Sw^,, Bw^ in equations (12), we have 

 ( /, mx^c'X ^ mxR) _ 



mZw= \ [Ic-'^^^ ]Ez-\-^—^\ St. 



And therefore 



X + iJi = 0, 





