486 G. H. KNIBBS AND F. W. BARFORD. 
principal value of Log —1 is t7i. Log —1is not, of course, . 
the same as log —1,* but by analogy we may take along 
the 2 axis the two points whose distances from the xy plane 
are -7. 
Lastly, since log —x = log x + log —1, we see that if 
= log x represents a curve in the «y-plane then it follows. 
that y = log —« may be considered to be represented by 
one of two curves, homothetic with the first, lying in planes 
which are parallel to the «y-plane, and passing through the 
two points on the z-axis defined as above. This may be 
considered as a particular case of representation by means 
of a bi-facial surface already discussed. It is evident that, 
with care, the use of logarithms of negative numbers pre- 
sents no insuperable difficulty. 
10. Sine curves.—The simplest and most general form of 
the sine curve is 
y = a, + a, sin (w + 0) +...... G,, sin m (#@ + OL)" eee (25) 
Given a series of equidistant values of y the method of 
solving for the constants d,......... dm, and for the epochal 
angles 9,,........ 6... is dealt with in many mathematical 
treatises. Where group-results are given for successive 
equal stretches of the abscissze, the necessary formule: 
have been deduced and given in a paper on the “ Statistical 
Applications of the Fourier Series.’ Solutions are given 
for groups up to the number 12. 
11. Parabolas and hyperbolas.—The general equation is. 
es (Ava Beet IAA (26) 
a parabola if m is positive, an hyperbola if m is negative. 
The solution is obvious if A = 0 for then 
log y = B tom log aor (27) 
+ For the difference between Log. - land log —1see Chrystal’s Algebra, 
Part 11, Ch. xx1x. 
2 G. H. Knibbs, c.m.e., ete. Journ. Roy. Soc. N.S.W., Vol. xiv, pp. 
76-110. 
‘a. 240 ee 
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