RECENT PROGRESS IN GEODESY. 
255 
exerted, the tangential strains find relief in the easiest pos¬ 
sible way. This is brought about by the transition from a 
sphere to the tetrahedron. When the earth contracts the 
same conditions exist, and the shrinking envelope assumes a 
figure giving a depression at the north pole and an elevation 
on, the continents of America, Europe, and Asia. This re¬ 
quires a depression between the two last continents which as 
a matter of fact existed in preglacial times. It seems to be 
a remarkable coincidence that the three arcs of parallels 
measured up to the present time should all agree in showing 
the curvature of the earth in their locality to be greater than 
the mean figure requires, and, as far as conclusions can be 
drawn from these results, we are forced to admit that the 
earth is collapsing in the form of a tetrahedron. One apex 
is found at the south pole, a base at the north pole, and the 
edges follow in general the continental upheavals. It may 
be worth while in passing to revert briefly to the geodetic 
measures necessary to determine the earth’s figure, and also 
to note the fact that in all this work the greatest attention 
has been given to the effect of plumb-line deflections on the 
length of the arc. The equations of Laplace, which exer¬ 
cise a control, and which also demand a consistency between 
the deflections in longitude and those in azimuth, have been 
steadily applied; so that there is little danger that the final 
arc as given in angular measurement is enough in error to 
invalidate our conclusions as to the real radius of curvature. 
The control is simply this: While we only have one method 
of determining the deflection in the plane of the meridian, 
there are two ways of doing this when it is required to find 
deflections in the plane of the parallel. It appears that the 
discrepancy between the astronomical and geodetic longi¬ 
tude, multiplied by a factor depending on the latitude of the 
place, is theoretically equal to the discrepancy in azimuth 
also multiplied by a function of the latitude. This furnishes 
a simple and effective check, and enables us to use all our 
astronomical observations in arriving at a consistent and 
most probable value of the great curve under consideration. 
