1912-13.] Fuller Test of the Law of Torsional Oscillation. 179 
an equation which gives discrete values of y corresponding to successive 
integral values of x. This condition also holds in the expression 
dy 
1+ ! (m 
i )y v 
P y m -\ 
k J ' 
where we explicitly make dy negative to suit the actual case. Hence, if 
three such successive points (y, x) lie nearly enough on a straight line, 
i.e. in integration, 
, 2 k 
log?/- — 
dy 27c dy _ - 2 ^ 
y pim - 1) y m ~ l m - 1 
1 
p (m — ] )(m - 2) 
— (m— 2) _ 
7.e. 
.m—2 
log y 
m — 1 
2 k 
(, x + a ), 
( m - !) 
pim — l)(m — 2) 
And if, as before, we write m = n + 2, this becomes 
2 / w | log y + — ———(x + a) 
[_ n + 1 
2Jc 
or 
pn{n + l) 
y n [ log y + k'(x + a)] = b . 
(5) 
This, then, is the new form of working equation, which differs from (4) 
simply in the introduction of the log y term. 
As before 
n log y + log [log y + k' (x + a)] = log b, 
so that the curve now to be used to give n is got by plotting log y against 
log [log y + k\x-{-a)\ But, as the value of k' is unknown, direct repre- 
sentation of results is impossible. 
In diagram 8,* showing the values of log y plotted against log (x + a), 
an s-shaped curve is obtained, which could, by choosing the two values of 
a — 0 and <x = 16, be changed to two straight lines, giving 71 = 3*8 and 
71 = 1*6 respectively. From this diagram the curve obtained when a = 5 
(an intermediate value) is chosen, and suitable values of y v y 2 , y 8 , x v x 2 , x 3 
are taken, so that a series of equations can be written, 
y{^%y 1 + k{x 1 + aj}=b 
y2 M [l°g2/2 + %2 + «)] = & 
2/3 W [l°§ Vz + k{x 3 + aj\ — b. 
Assume, for convenience 
2/ 2 = x 2/i 
2/ 3 = a 2/ 2 
* Proc. Roy. Soc. Edin ., vol. xxxi. p. 436, 1910-11. 
