678 PROFESSOR TAIT ON 
with the notation of (2). Now, however the forces may turn, 
$8 = ZaSBB 
is an absolute constant ; for each scalar factor as S8,6 is unaltered by rotation. 
Let us therefore change the origin, ¢.¢., the value of each a, so as to make 
LTaSBC=0, 
Lels, . ¢8=0 J i ll ee oe (6) 
Thus we see that ¢ may be expressed”in the form given in (4’) above. 
4, Equation (5) is now 
bVBp=Vyy' + V80,, ‘ P : : (5') 
where 0 is the tensor, and £ the versor, of 8. 
The condition that the force shall lie in thefplane of the couple is, of course, 
included in this, and is found by operating by 8.6. Thus 
SS/sy8)S0 2. 2. 
We have here all the data of the problem, and solutions can only differ from 
one another in the. mode of attacking (5’) and (7). The most purely quater- 
nionic mode, so far as Hamitron developed his calculus, seems to be the 
following :— 
Writing (7) in the form 
Sy(o'+ VBy’)=0, 
we have at once 
ty = VBS + BVBy, f: 
whence t6= —VBy' + BVBS’, 
where ¢ is an undetermined scalar. 
By means of these we may put (5’) in the form 
btVBp=V.B(Vy'S +7'SyB+8'S8'B) 
=V.B(V7's —o8) 
where 
w=—ySy( )—5S8( ). 
