94 
Proceedings of the Royal Society of Edinburgh. [Sess. 
As already mentioned, it will appear from the observations of resistance, 
to be given subsequently, that the resistance may be taken as varying 
with the first and third powers of the parameter x and not with the second. 
The same is true of circular error, due to the consideration of a finite arc 
of oscillation. Confining attention then to actual cases, consider 
0 = x" + kx + n 2 x + A^ 3 + X 2 x 2 x + \„xx 2 + A 4 F 3 + . . . (8a) 
where accents denote differentiations. 
Substitute 
X = a sin (n't + e 0 ) = a sin r, 
then 
x 3 = a 3 sin 3 T 
= fa 3 sin r — fa 3 sin 3 r 
3 ,, 2 ,y, 
+ • 
X X 
.2 
= no? cos r sin 2 r = 4 n a? cos r — 4Fa 3 cos 3 r = 4a 2 F + . . . 
x' 2 x = n 2 a 3 cos 2 r sin r = \n 2 a 3 sin r + \n' 2 a 3 sin 3 t = \ri 2 a?x + . . . 
x 3 = n' 3 a 3 cos 3 t 
= \n 3 a 3 cos T + \n 3 a 3 COS 3 r = %n 2 a 2 x + . 
It will appear below that terms such as sin3T, cos 3 t are without any 
sensible influence ; hence they are disregarded ; the equation becomes 
0 = x" + [k + i A 2 a 2 + |'/V 4 7z' 2 a 2 ]x / + [ n 2 + f A 4 a 2 + JA 3 ^ 2 a 2 ] 
x 
showing how period, etc., are modified by higher terms. 
For example, if we have as the more exact equation 
in place of 
0 = x" + . . . + n 2 sin x 
0 = x" + . . . + n 2 x, 
(86) 
the correction is given by Apr 3 = n 2 (sin x — x) = — \n 2 x 3 , and that to the 
frequency is in consequence 
An 
n 
= +§Ai a2 = 
- -i-a 2 
16 a 
which is the Circular Error, as usually given. 
Further treatment of this point is given in the Tables of Circular Error, 
which will appear hereafter. 
With regard to the terms in sin 3 r, . . . which have been passed over : 
suppose they were included, and we wrote 
x" + . . . + [ n 2 -f . . . ]x = -JA 1 a 3 sin 3r + . . . 
we should have in consequence 
-gV A x a hi 2 sin 3 r . . . 
Taking the same example, with \ = — \n 2 , the coefficient of the term 
would be 
x— . . . — ^a 3 sin 3 t+ ; 
