484 G. H. KNIBBS AND F. W. BARFORD. 
The values of logarithms, therefore, of negative numbers 
may be regarded as numerically equal to those of positive 
numbers, but inverted in reference to the space occupied by 
the positive numbers.* Hence the values, which may be 
differentiated by prefixing « thereto are not continuous with 
the logarithms of positive numbers, in an analogous way 
to that in which the values of y = }/« are regarded as 
imaginary when the values of x are negative. 
9. Geometrical conventions for representing the loga- 
rithms of negative numbers.—The matter may be looked 
at in two other ways. In Fig. 2 the ‘‘ normal curve ”’ 
represents the values of the logarithms of + 2, and the 
‘‘inverted curve”’ the values of the logarithms of —x. The 
latter may be regarded as an inverted image of the former. 
There is still another way of regarding the representa- 
tion of log -1. 
Let « denote the operator which, when applied to log 1 
converts it into log —1. If the operator « is again applied 
to log —1, we get log —1 or wlog1. If the nature of 
the operator + is such that «. = 1 we are brought back 
again to our starting point which was log 1.’ 
1 Mobius was, we believe, the first to recognise that a simple type of 
surface can be unifacial and unimarginal. It has been suggested that 
the ordinary plane of projective geometry is unifacial. See Klein, Math. 
Annalen, Bd. vu, p. 549. 
2 wis not ce? but the operation of « upon uv, that is the operation twice 
repeated. 
