480 G. H. KNIBBS AND F. W. BARFORD. 
Hence, writing Ys for the left-hand member and taking 
logarithms 
if lor, Ve, (14) 
ali MEE oe 
ak log,e = log, V3, ora = 
in which M is 2°30258509...... 
When a is found the solutions for C and B are obvious. 
Should the left hand member of (13) be negative this curve 
is unsuitable. 
It may be noted that, in general, if a quantity to be 
operated upon is not in the form of a product or a power it 
should be converted if possible into such a form. For 
example, in the equation 
y= = 8 cos’a — 8 cos*a + 1 = cos 4a, eee (15) 
the initial and final terms give 
ta meet ATO (16) 
log y 
though from the intermediate terms a solution cannot be 
directly obtained. 
In expressions like (8) « may indifferently + or —, butin 
Uy Ne Sener oe, eee (17) 
care has to be taken in respect to the interpretation of the 
logarithm of negative numbers; see §8 hereinafter. 
7. The antilogarithmic generatrix.—What has been some- 
times called the antilogarithm of a number, is that number 
the logarithm of which is the number in question. It is 
convenient, following the analogy of inverse trigonometric 
functions, to express this operation by prefixing log—! or 
A-* to the number. Thus, if log y = 7, then log—*7 or 
A—"y = y. Thus, if log f («) = 7, the curve y = / (#) may 
be called its first anti-logarithmic generatrix. Y¥or 
example, if 
Y = -PomM 20.2 ed UB) 
these quantities actually being logarithms, then we have 
A—ty=A—* (a+ mr) =(A—*a) (AT me) = Y= AL" 
where log Y = y; log A =a; log X = x; log M = m. 
