162 EF. LANCASTER JONES 
sections of the path is equal to the mean of the values at the two sta- 
tions connected by it, and taking the sum of the sectional differences 
of gravity thus computed. Thus, if P and Q are any two consecutive 
stations on the path chosen, and if their coordinates are (21, y1) 
and (x2, y2) whilst their component gravity-gradients are (X1, V1) 
and (Xe, Ye) respectively, we have 
2(gp — ga) = 4 [(X1 + X2)(%2 — a1) + (Vi + Yo)(y2 — y)]. 
If, however, we take another path connecting A to B and involving 
other intervening stations, we shall usually obtain a different value 
for ga—ga. 
This is a particular case of a more general problem concerned with 
single-valued functions, and it is with this general class that we are 
concerned. 
§2. THE PROBLEM 
Given a network of stations A, B, C,--- and the observed or 
computed values of the increments (ad), (dc), (ca), - - - between ad- 
jacent stations of a single-valued function U, we have to find the 
best possible values of U at A, B,C,---. 
The problem resolves itself into one which may be expressed 
thus: To find the adjusted values (AB), (BC), (CA), - - - of the incre- 
ments (ab), (bc), (ca), ---, respectively, where, around any closed 
polygonal path ABC - - - PA, we have the condition that 
(CAB BO) (GD) Al) ras 
whereas, in general 
(ab) + (bc) + (cd) + --- + (pa) + o but = some value d. 
We may call d the “excess” around the path ABC --- PA. 
In the particular cases of gravitational and magnetic observations 
in applied geophysics, this problem and its solution by the method of 
least squares have been considered by I. Roman.? Roman states the 
problem in a different but equivalent manner and obtains rather more 
complex normal equations, which he solves by the usual reduction 
method of Gauss. 
The present writer believes that the treatment given below is 
particularly adapted to this problem. The normal equations are simple 
and can be written down by inspection. Methods of resolution of 
these equations are suggested which are much less laborious than the 
Gaussian reduction methods, and permit the labour of reduction to 
* Trans. Amer. Inst. Min. Metall. Engrs., Geophysical Prospecting, 1932, p. 460. 
822 
