612 ON THE FIGURE, DIMENSIONS, AND MEAN SPECIFIC GRAVITY OF THE EARTH, 
Society’s Scale and the Ordnance Standard O2, it was determined that 
Ord. Standard Oj= 1 19 997508 mean inches of the centre yard of the Royal Astronomical Society’s Scale ; 
also from Mr. Baily’s comparison of this scale with the standard metre he deter- 
mined* 
Standard Metre =39'369678 mean inches of the centre yard of the Royal Astronomical Society’s Scale. 
From more recent observations, it appears that the Royal Astronomical Society’s 
scale has undergone a permanent alteration of length ; the interval however between 
the two series of observations above quoted was not sufficiently long to vitiate the 
connexion thus established between the Ordnance Standard and the metre. The 
resulting value of the metre in terms of O, is therefore 
Standard Metre =3*2808746 mean feet of O,, 
and hence since the metre =443*296 lines of the toise of Peru, 
Toise =6*3945438 mean feet of O,. 
Reduction of the Triangulation. 
If u represent the true ratio of the distance between any two points in a network 
of triangulation to the base line ; A B C...A'... the true angles, whose observed values 
are A-j-a, B-}-j3, C-l-7.*..A'-j-a'..., then if w, be the calculated value of u obtained by 
using the series of observed angles ABC..., the value obtained by using the series 
of observed angles A'B'C' 
u =/( ABC. ..)=/(A'B'C'. ..) = .... 
u^=.u-\-a a -\-b^ +cy +.... 
M j M -|- a! a' -j- 6'/3' -j- c'y’ -j- 
and so on. Each different calculation of u will therefore give a different value for 
that quantity. 
In the necessary existence of these discrepancies among the calculated values of 
u, it becomes of much importance to obtain the most probable value. In ordinary 
calculations this has been generally effected by assuming it to be the mean of all the 
calculated values u{ii{u[ This might be improved upon by assigning to each value 
of u its proper weight by means of the weights of the observed angles, but the method 
would still be imperfect and discrepancies would still exist in other parts of the 
work. 
From the above equations, though we cannot determine the precise value of u, yet 
we can obtain some precise information respecting the errors of observation ; for we 
have evidently, since the quantities ahc...a'b'c' ... are numerical, certain equations of 
condition between the unknown errors. 
But the number of such equations of condition for the whole figure being necessarily 
less than the actual number of errors, an indefinite number of systems of corrections 
* Memoirs of the Royal Astronomical Society, vol. ix. 
