John Henry Pratt (c. 1811-1871), English clergyman and
mathematician, 'spent many years in India as archdeacon of Calcutta.
His observations and deductions laid the foundation for the later
development of the principle of isostasy.
It is now well known that the attraction of the Himalaya
Mountains, and of the elevated regions lying beyond them, has a
sensible influence upon the plumb-line in North India. This
circumstance has been brought to light during the progress of the
great trigonometrical survey of that country. It has been found by
triangulation that the difference of latitude between the two extreme
stations of the northern division of the arc, that is, between
Kalianpur and Kaliana, is 523'42".294, whereas astronomical
observations show a difference of 523'37".058, which is 5".236* less
than the former.
That the geodetic operations are not in fault appears from this;
that two bases, about seven miles long, at the extremities of the arc
having been measured with the utmost care, and also the length of the
northern base having been computed from the measured length of the
southern one, through a chain of triangles stretching along the whole
arc, about 370 miles in extent, the difference between the measured
and the computed lengths of the northern base was only 0.6 of a foot,
an error which would produce, even if wholly lying in the meridian, a
difference of latitude no greater than 0".006.
The difference 5".236 must therefore be attributed to some other
cause than error in the geodetic operations.
This is the difference as stated by Colonel Everest in his work on
the Measurement of the Meridional Arc of India, published in 1847.
See p.clxxviii.
A very probable cause is the attraction of the superficial matter
which lies in such abundance on the north of the Indian arc. This
disturbing cause acts in the right direction; for the tendency of the
mountain mass must be to draw the lead of the plumb-line at the
northern extremity of the arc more to the north than at the southern
extremity, which is further removed from the attracting mass. Hence
the effect of the attraction will be to lessen the difference of
latitude, which is the effect observed. Whether this cause will
account for the error in the difference of latitude in quantity,
as well as in direction, remains to be considered, and is the
question I propose to discuss in the present paper.
To dissect and actually to calculate the attraction of the masses of
which the Himalayas, and the regions beyond, are composed, appears,
at the very thought of it, to be an herculean undertaking next to
impossible. I am fully convinced, however, that no other method will
succeed. It is upon this plan that the solution of the problem is
conducted in this paper. It will be seen, that by selecting a
peculiar law of dissection the calculation is very greatly
simplified, and made to depend entirely and solely upon a knowledge
of the elevations and depressions, in fact, the general contour of
the surface. This information for some part of the mass is already
supplied by the maps of the Trigonometrical Survey.
In the following pages I propose, in the first place, to develop
my method of calculation, and to deduce a formula by which the
attraction can be determined with a precision corresponding to the
degree of accuracy to which the contour of the surface is
known.
In the second place, I propose to reduce the formula to numbers,
and so arrive at such an approximate value of the attraction as the
data I have been able to collect will allow.
This approximate value is, as will be seen, larger than 5".236,
the error brought to light by the Survey. I make various suppositions
with a view, if possible, to reduce my result to this, but without
effect. This leads me to look in another direction for an explanation
of the cause of discordance, and I arrive at a conclusion which
clears up the discrepancy, confirms the calculations of this paper,
and illustrates the importance of not disregarding the influence of
mountain attraction.
Adding together the results of the last article, we have
Deflection of plumb-line in meridian at A . . . = 27".853
Deflection of plumb-line in meridian at B . . . = 11".968
Deflection of plumb-line in meridian at C . . . =
6".909
. .Difference of meridian deflections at A and B = 15".885
Difference of meridian deflections at A and C = 20".944
Difference of meridian deflections at B and C 2
5~.059
The quantities is first of these considerably greater than
5".236, the quantity brought to light by the
Indian Survey. And the values of the
deflections at B and C bear a far higher ratio to those at A than has
been generally supposed....
The conclusion, then, to which I come is, that
there is no way of reconciling the difference
between the error in latitude deduced in
Colonel Everest's work and the amount I
have assigned to the deflection of the plumb-line arising
from attraction--and which, after careful re examination, I am
decidedly of opinion is not far from the truth, either in defect or
excess--but by supposing, that the ellipticity which
Colonel Everest uses in his calculations, although correct as a mean
of the whole quadrant, is too large for the Indian arc. This
hypothesis appears to account for the difference most satisfactorily.
The whole subject, however, deserves careful examination; as no
anomaly should, if possible, remain unexplained in a work conducted
with such care, labour, and ability, as the measurement of the Indian
arc has exhibited.
The Astronomer Royal in a paper published in the Transactions for
1855, suggested that immediately beneath the mountain-mass there was
most probably a deficiency of matter, which would produce, as it
were, a negative attraction and so counteract the effect of the
plumb-line. This hypothesis appears, however, to be untenable for
three reasons:--(1) It supposes the thickness of the earth's solid
crust to be considerably smaller than that assigned by the only
satisfactory physical calculations made on the subject --those by Mr.
Hopkins of Cambridge. He considers the thickness to be about 800 or
1000 miles at least. (2) It assumes that this thin crust is lighter
than the fluid on which it is supposed to rest. But we should expect
that in becoming solid from the fluid state, it would contract by the
loss of heat and become heavier. (3) The same reasoning by which Mr.
Airy makes it appear that every protuberance outside this thin crust
must be accompanied by a protuberance inside, down into the fluid
mass, would equally prove that wherever there was a hollow, as in
deep seas, in the outward surface, there must be one also in the
inner surface of the crust corresponding to it; thus leading to a law
of varying thickness which no process of cooling could have
produced.
It is nevertheless to this source--I mean a Deficiency of Matter
below--that we must look, I feel fully assured, for a compensatory
cause, if any is to be found. My present object is to propose another
hypothesis regarding the deficiency of matter below the
mountain-mass, as first suggested by Mr. Airy; and to reduce my
hypothesis to the test of calculation....
I will now state the hypothesis on which my present calculation
proceeds. At the time when the earth had just ceased to be u holly
fluid, the form must have been a perfect spheroid, with no mountains
and valleys nor ocean-hollows. As the crust formed, and grew
continually thicker, contractions and expansions may have taken place
in any of its parts, so as to depress and elevate the corresponding
portions of the surface. If these changes took place chiefly in a
vertical direction, then at any epoch a vertical line drawn down to a
sufficient depth from any place in the surface will pass through a
mass of matter which has remained the same in amount all through the
changes. By the process of expansion the mountains have been forced
up, and the mass thus raised above the level has produced a
corresponding attenuation of matter below. This attenuation
is most likely very trifling, as it probably exists through a great
depth. Whether this cause will produce a sufficient amount of
compensation can be determined only by submitting it to calculation,
which I proceed to do.
The Tables thus calculated furnish the following
results:--
|
|
At Kaliana |
At Kalianpur |
At Damargida | |
|
Deflections in meridian, caused by a mass beyond of the Himmalayas and the Mountain region |
27".978 |
12".047 |
6".790 | |
|
Ditto, by the same mass distributed through a depth of 100 miles |
26.440 |
12.111 |
6.855 | |
|
Ditto |
300 miles |
21.106 |
11.678 |
6.866 |
|
Ditto |
500 miles |
17.066 |
9.622 |
6.670 |
|
Ditto |
1000 miles |
11.199 |
7.386 |
5.220 |
From Proceedings of the Royal Society of London.
Vol.Xiii, pp. 253-276, 1864
In fact the density of the crust beneath the
mountains must be less than that below the plains,
and still less than that below the ocean-bed. If solidification from
the fluid state commenced at the surface, the amount of contraction
in the solid parts beneath the mountain-region has been less than in
the parts beneath the sea. In fact, it
is this unequal contraction which appears to have caused the
hollows in the external surface which have become the basins into
which the waters have flowed to form the ocean. As the waters flowed
into the hollow thus created, the pressure on the ocean-bed
would be increased, and the crust, so long as it was
sufficiently thin to be influenced by hydrostatic principles of
floatation, would so adjust itself that the pressure on any coucbe
de niveau of the fluid should remain the same. At the time
that the crust first became sufficiently thick to resist fracture
under the strain produced by a change in its density--that is, when
it first ceased to depend for the elevation or depression of its
several parts upon the principles of floatation, the total amount of
matter In any vertical prism, drawn down into the
fluid below to a given distance from the earth's centre, had been the
same through all the previous changes. After this, any further
contraction or any expansion in the solid crust would not alter the
amount of matter in the vertical prism, except where there was an
ocean; in the case of greater contraction under an ocean than
elsewhere, the ocean would become deeper and the amount of matter
greater, and in case of a less contraction or of an expansion of the
crust under an ocean, the ocean would become shallower, or the amount
of matter In the vertical prism less than before. It is not likely
that expansion and contraction in the solid crust would effect the
arrangement of matter in any other way. That changes of level do take
place, by the rising and sinking of the surface, is a
well-established fact, which rather favours these theoretical
considerations. But they receive, I think, great support from the
other fact, that the large effect of the ocean at Punnoe and of the
mountains at Kaliana almost entirely disappear from the resultant
defections brought out by the calculations.
This theory, that the wide ocean has been collected on parts of the
earth's surface where hollows have been made by the contraction and
therefore increased density of the crust below, is well illustrated
by the existence of a whole hemisphere of water, of which New Zealand
is the pole, in stable equilibrium. Were the crust beneath only of
the same density as that beneath the surrounding continents, the
water would be drawn off by attraction and not allowed to stand in
the undisturbed position it now occupies.
I have, in what goes before, supposed that, in solidifying, the
crust contracts and grows denser, as this appears to be most natural,
though, after the solid mass is formed, it may either expand or
contract, according as an accession or diminution of heat may take
place. If, however, in the process of solidifying, the mass becomes
lighter, the same conclusion will follow--the mountains being formed
by a greater degree of expansion of the crust beneath them, and not
by a less contraction, than in the other parts of the crust. It may
seem at first difficult to conceive how a crust could be formed at
all, if in the act of solidification it becomes heavier than the
fluid on which it rests; for the equilibrium of the heavy crust
floating on a lighter liquid would be unstable, and the crust would
sooner or later be broken through, and would sink down into the
fluid, which would overflow it. If, however, this process went on
perpetually, the descending crust, which was originally formed by a
loss of heat radiated from the surface into space, would reduce the
heat of the fluid into which it sank, and after a time a thicker
crust would be formed than before, and the difficulty of its being
broken through would become greater every time a new one was
formed.
The least that can be gathered from the deflections of these
coast-stations is, that they present no obstacle to the theory so
remarkably suggested by the facts brought to light in India, viz.
that mountain-regions and oceans on a large scale have been produced
by the contraction of the materials, as the surface of the earth has
passed from a fluid state to a condition of solidity-- the amount of
contraction beneath the mountain-region having been less than that
beneath the ordinary surface, and still less than that beneath the
ocean-bed, by which process the hollows have been produced into which
the ocean has flowed. In fact the testimony of these coast-stations
is in some degree directly in favour of the theory, as they seem to
indicate, by excess of attraction towards the sea, that the
contraction of the crust beneath the ocean has gone on increasing in
some instances still further since the crust became too thick to be
influenced by the principles of floatation, and that an additional
flow of water into the increasing hollow has increased the amount of
attraction upon stations on its