478 DR G. PLARR ON THE DETERMINATION OF 



or grouping the terms in z 2 together 



Y' = u\ - 1 - b 2 u 2 + b 2 (a 2 + 2b 2 )] 



+ u*z 2 [2-mb 2 (t + L +L 1 )] . 



and as 



put this=L 2 . 

 Thus we have, 



a 2 + 2b 2 = L : — mb 2 = -7?r„ 

 a~b- 



+^[2a 2 b 2 + (t + U+^]. 



2a 2 b 2 + L = 2a 2 b 2 + b 2 a 2 + b 2 u 2 

 L^w 2 ^ 2 -^) 

 2a 2 b 2 +L + L 1 = Sa 2 b 2 +u 2 (b 2 -U 1 ) + 6u\ 



Y' = u 2 [b 2 (l x -u 2 )-q 

 Z' = u 2 [6 2 (^-u 2 )-^] 



where 



L 2 = 3a 2 & 2 -u 2 (4a 2 +76 2 ) + 6u 4 . 

 At the end of this paper we will show that we have identically 



M 1 _M 1 '_M/'_ 



— — xN j 



1 9 3 



where c^, a 2 , a 3 are the scalar coffiecients of a decomposed parallel to i, j, k, so that 



a = ia, +ja 2 + Jca 3 . 

 The equations M/ = 0, M/' = give therefore no new relations besides those which we 

 have deducted from (1), (2), (3). We may incidentally also remark here, that by the 

 geometrical nature of the question we have identically 



because the point of contact (or of tangence) cannot change when the rotation-axis 

 coincides with the axis of figure. 



We remark that Y', 71 are of the forms 



Y^Xo+3%! jX = u 2 [fc 2 (Z x -u 2 )-/] 



Z^Xo + ^j-^ 6 [x x =^(L 2 +t) s 



and remembering 



Y 2 = (u 2 y 2 -b i )(u 2 -b 2 ) 

 Z 2 = (u 2 z 2 -b i )(w 2 -b 2 ), 

 we form the two combinations 



(III.) y 2 Y' 2 + z 2 Z' 2 - ( Y 2 + Z 2 )R' 2 = 



(IV.) 2/ 2 Y' 2 -z 2 Z' 2 -(Y 2 -Z 2 )R' 2 = 



