NAPIER COMMEMORATIVE LECTURE. 57 
Let the common initial velocity be er. 
Then « = ert and AG — y) = cy. 
From the second of these equations ou + cy = 0. 
Thus, using the initial conditions, y = re—“. 
Therefore x = r log. ; 
If we write logny for Napier’s logarithm of y, as defined 
aes x = logny =r log. : 
The relation between x and y canalso be put in the form 
y=10' e~in, when r = 10°. 
Pe. yaiw | no G-2) 
| 
\Mm—> @ 
On comparing this with 
1 10% fior 
y = 10") (1- 107? ’ 
we see that the values of y given by the latter will not 
differ by much from those given by the former. 
From the relation « = logyy =r loge, , itisclear that — 
if points are taken on AB such that BPi, BP2, BPs, etc., 
descend in G.P., the corresponding points on CD, namely 
Q1, Q2, Q3, etc., ascend in A.P. 
Further, the logarithm of radius is zero. 
Also logy(u v) is not logyu + logyv, neither is logy(u/v) 
the same as logyu — lognv. 
But if z = = lognz = logyu + logyv, 
and if “= a lognz = lognu — lognv. 
Finally, if ma _ A , then logyu — logyv = logyw’ —lognv’. 
