482 G. H. KNIBBS AND F. W. BARFORD. 
For negative values of y it may be said that log y or 7 is 
an impossible quantity, and therefore there can be no 
logarithmic homologue. Or again, if « be negative, there 
is similarly no logarithmic homologue since log x or € is an 
impossible quantity. We can, however, conceive the 
matter thus:—Let us first suppose that the value of y is 
positive. Thus in Fig. 1 shewing 7 = n&, we have, if we 
plot the points P for various values of é, n the tangent of 
6, or the angle of intersection of the line passing through 
the points and the axis Op=. The point P of the logarith- 
mic homologue corresponding to the value x, moves from P 
to O as « changes from + © to +1; from O to P’asx 
changes from +1 to +0. As « changes from —0to —1, 
P’ moves on the inverted face of the same surface from P’ to 
O, and finally as « changes from —1 to — ©, P’ moves on the 
inverted face of the line OP. Or representing the result 
on an infinitely great sphere—See Fig. la—we can call the 
face POP’ the normal, and P'O’P the inverted face, (reach-~ 
ing the paradox that the logarithm of — © becomes the 
same as that of +o) which is not only a matter of no 
moment but a difficulty that arises in other schemes of 
curve tracing. 
Secondly, let us suppose that y is negative, that is, that 
—y=2. Then we have y = —(«") and the representation 
isas before. Or,again, we may suppose the representation 
to start at O' and it will be as shewn by fine dotted lines 
on Figs. 1 and 1a. 
