554 TRANSACTIONS OF SECTION E. 
formula, the Principal Triangulation of the United Kingdom appeared to be 
inferior in accuracy to the Continental systems. Various proposals for the 
remeasurement of the principal ares were made, e.g., at the 1906 and 1908 
meetings of the British Association for the Advancement of Science at York and 
Dublin. In 1908 it was decided to remeasure a smail portion of the Principal 
Triangulation remote from the bases in order to ascertain what linear errors 
had accumulated, and whether a whole remeasurement was advisable. This 
partial remeasurement was carried out in the years 1909-12, and included the 
measurement of a new base at Lossiemouth on the Moray Firth, and a small 
network of triangulation to connect this base to three of the original stations 
of the Principal Triangulation. 
The result of this operation shows an error in the published values of the 
lengths of the sides of about one inch in one mile (1/60,000) in Morayshire. 
We have now, therefore, four tests on the linear accuracy of the Principal 
Triangulation. 
1. The three independently measured base lines at Lough Foyle, Salisbury 
Plain, and Lossiemouth. 
2. The side Cassel-Les Harlettes of the French Meridional Arc, connected to 
the Principal Triangulation of the United Kingdom in 1861-62, and depend- 
ing on the Paris Base (remeasured 1890). 
The following table shows the comparison between the measured lengths of 
these bases and the same lengths as calculated through the Principal Triangula- 
tion from any other of the bases :— 
Tasre 1. 
The accordance of the bases as shown through the triangulation. (A + sign 
indicates that the calculated lengths of the bases in column 1 are greater than 
the measured.) Resulting differences are shown for each case both in units of 
the seventh place of logarithms and as a fraction. 
Salisbury Plain | Lough Foyle | Lossiemouth Paris 
Bases ,a8 a8 Las .fe 
“ Log. | 2 = = | Log. | 23:5 Log. | 2 5 Log. | 2 3 
Dif. | $838 | Diff. | 68s]. Diff. |SSs| Diff. | 68s 
o Be O Fe C ar O Fe 
lisb Plai 46°5 = 95°8 sith 8-7 sani0 
Salisbury Plain eo Bas weed 93000 |— 45000 | ; 
1 a 1 _——s 
Lough Foyle . | + 46°5 93000 | vr |— 49°38) ge 009 | + 55°2 ici 
1 : 1 ae 
Lossiemouth . | + 95:8 5000 +49°3 88000 «-» |+104:5) 42000 
2 1 1 1 
Paris . . —87 505000 — 55:2 79000 — 104°5 42000 
If bases are regarded as errorless relatively to triangulation connecting 
them, the discrepancies between the measured lengths of bases and those same 
lengths calculated from adjacent bases will afford a method of comparing the 
telative precision of different systems. 
‘Throughout a system homogeneous in size and shape of figure, and in pre- 
cision of angular measurement, the linear errors generated would vary as the 
square root of the distance from the base. If, therefore, we refer to lists of 
the discrepancies between the measured and calculated lengths of bases in other 
National Triangulations and multiply each discrepancy by the factor WS = 
(where d is the distance in miles between the bases), we get a table which 
will serve to give a rough idea of the relative precisions of the various systems, 
and of what errors we should expect to find generated in them at a distance 
of 100 miles from the original base. 
