28 
Proceedings of Royal Society of Edinburgh. [sess. 
corresponding magnitudes that constitute Q and Q\ the “three 
imaginary units 55 i, j, h must of necessity fulfil Hamilton’s equa- 
tions — 
i 2 = 
ij = h, jjc = i, hi = j 
ji — ~h, hj= -i, ih = -y. 
In like manner the product of two vectors 
vu = (ix 4 -j y + kz)(ix +jy' + kz) 
leads to Hamilton’s well-known scalar and vector products ; and 
the usual geometrical meanings of these are given with reference 
to the vectors which enter into them. 
Thus, according to Klein and Sommerfeld, i. j , h are vectors as 
well as imaginary units; and they are also regarded as Wende- 
streckungen of tensor unity (p. 61), that is, as operators producing 
a semi-revolution ( Umklappung) round an axis. They say : — 
“The resultant of two semi-revolutions about the same axis is 
identity ; two semi-revolutions about mutually perpendicular axes 
give a semi-revolution about the normal to the two axes. If 
we wish to make the algebraic sign right, we must, as on p. 36 and 
following, pass from the consideration of the whole to that of the 
half angle of rotation. Then we recognise: it will be i 2 = — 1, 
because i 2 has to do with a whole revolution, whose half angle of 
rotation to modulus 27r is equal to tt (and not equal to zero). 
Moreover, the formulae (8) \i 2 =j 2 = It 1 = -1] recall the equation 
i 2 = - 1 in the theory of ordinary complex numbers.” 
The reference to p. 36 is simply a reminder that the expressions 
A, B, C, D , involve sin — and cos and not sin w and cos w, and 
! 2 2 
that there are difficulties in regard to the signs. 
But if, in any true symbolic sense, i is to represent a semi- 
revolution about an axis, and if, following Klein and Sommerf eld’s 
notation, we represent the semi-revolution of the body B about the 
2 -axis by the symbol B/, have we not good reason to expect that 
B ii should be equal to B, i.e., i 2 = + 1 ? Klein and Sommerfeld 
say distinctly that B i 2 is identical with B ; and yet i 2 is also to 
be equal to - 1, because of half angle considerations and the 
theory of complex numbers! This “facing both ways” of i 2 
