266 PROFESSOR TAIT ON THE ROTATION OF A 



as we might at once have seen by the geometry of the question. 

 Hence 



q = x + yV(a - aj) (/3 - /3J . 



By the help of this, the first of equations (3) becomes 



= x(a-a 1 ) + y {V(a-a 1 )(/3^/3J.. a -a 1 V{a-« I )t0-/3 1 )} 



or 



= .X + 7/S(a+a 1 )(/3-^ 1 ). 



[The second of equations (3) merely gives us a condition which is equivalent 



to this, because 



S(«+ «J (/3-/3,) =-S(«- ■ 1 )(/S + j8 1 ) 

 or 



Sa/3 = Sa 1 /3 1 .] 



Thus, finally, 



r / = y(-S( a -ra 1 )(/3-/3 1 ) + V(« - aj ((3 - Pj) 

 = -,y[(/3-^ 1 ) a + ai (/3-/3 1 )] 



where, as was to be expected, the tensor is left indeterminate. 



7. Given the instantaneous axis in terms of the time, it is required to find the 

 single rotation which will bring the body from any initial position to its position at 

 a given time. 



If a be the initial vector of a point of the body, ™ the value of the same at 

 time t, and q the required quaternion, we have 



ar =5 quq— 1 (4). 



Differentiating with respect to t, this gives 



zr = quq— 1 — q<*-q~ l qq~ l ' 



= il ~ 1 • 2 a 2 —1 — (7 a <Z — X • 9.1 ~ X > 

 = 2V . (Vqq- 1 . quq' 1 ) . 

 But ®" = Vezr = V . sqaq- 1 . 



Hence, as qaq~ x may be any vector whatever in the displaced body, we must 

 have 



e = 2Yqq~ 1 (5). 



This is the fundamental kinematical relation already referred to. Cayley's* 

 quaternion form of it (which will be understood by the help of § 13 below) is 



, . . , , ~ dA dx. 



* Phil Mag., Sept. 1848. 



