238 Proceedings of the Royal Society of Edinburgh. [Sess. 
The standard S.H.M. for comparison with the actual motion is one of 
the same phase and period, having its range in the line CQ and conter- 
minous with that of the pen.* The magnitude of the range is 
2V[2^ 2 + l + 2p]-2 x /[2^ + l-2p] ... (3) 
The actual position of the pen = QP 
= -1- 1 + 2p (sin $ — cos 0 ) + Vl^P 2 + 1 - 2p(sin 0 + cos 0)], 
2 e 
and its true position 
= {\/[ 2 P 2 +l + 2p]+ V[2p 2 + 1 -2^]} -cos 6 { J [ ]- V[ ]} 
Hence 
f \ fl 2 P 2 +1+22? ( sin e ~ cos 9 ) + J [' 2 P 2 + 1-2^? (sin 9 + cos 0)] ~ { \/C2 p 2 + 1 + 2^?] + >/'[2p 2 + 1 - 22?]} ) +cqc , 
\ v /[22? 2 + 1 + 22?] - J[2p* + 1-22?] / 
Table of the Error as a Function of the Dimensions of the 
Duplex Unit and the Phase Angle. 
The values of e have been calculated for p = 1, 4, 5, and 10; and for 
0 = 0 °, 15°, 30°, 45°, 60°, 75°, 90°, 105°, 120°, 135°, 165°, and 180°. 
As the quantity to be calculated involves the difference of the positive 
and negative parts of the numerator of the fraction, and these are nearly 
equal in magnitude, it was found necessary to use 7-figure logarithms. 
Table II. 
p = l . 
0 1 
0 ° 
15 
30 
45 
60 
75 
90 
105 
120 
135 
150 
165 
180 
6 
0 
-•0154 
- -0575 
- -0873 
- -0682 
- -0296 
0 
+ •0165 
•0186 
•0147 
•0079 
•0022 
0 
P = 
e 
= 4 . 
0 ° 
15 
30 
45 
60 . 
75 
90 
105 
120 
135 
150 
165 
180 
e 
0 
- *0007 
- -0022 
- -0034 
- -0034 
- -0020 
0 
+ •0017 
•0025 
•0022 
•0013 
•0004 
0 
P = 
9 1 
= 5 
0 ° 
15 
30 
45 
60 
75 
90 
105 
120 
135 
150 
165 
180 
. 1 
0 
- *00041 
- -00134 
- -00210 
-•00211 
- -00128 
0 
•00112 
•00164 
•00143 
•00087 
•00027 
0 
P = 
d 
= 10 . 
1 0 ° 
15 
30 
45 
60 
75 
90 
105 
120 
135 
150 
165 
180 
e 
1 0 
- -00009 
- -00031 
- -00048 
- -00049 
- -00032 
0 
•00030 
•00044 
•00042 
•00025 
•00008 
0 
These results are graphed in fig. 8. (The curve for p = 1 has been 
drawn to the vertical scale of the other curves in order that it may be 
* This is not that particular S.H.M. to which the actual motion is most akin, but it is 
sufficiently nearly so ; and it is far the most convenient both for the user of the instrument 
and in the calculation of the error. For further information on this point see the previous 
paper. * 
