TRANSACTIONS OF THE SECTIONS. 8 
‘Theorém Fifth.—For each axis of elasticity of a perfect solid, the coefficient of lon- 
gitudinal elasticity is equal to three times the sum of the two coefficients of rigidity 
for the coordinate planes which pass through that axis, diminished by three times 
the coefficient of rigidity for the plane normal to that axis. 
Thus in perfect solids all the coefficients of elasticity are functions of three inde- 
pendent coefficients—those of rigidity. In no previous investigation has the num- 
ber of independent coefficients been reduced below siz. 
To represent the phznbdiiena of imperfect solids, there is introduced the hypothesis 
of molecular vortices in addition to that of atomic cehtres ; that is to say, each atomic 
centre is supposed to be surrounded by a fluid atmosphere, retained round the centre 
by attraction, and diffused from it by the centrifugal force of revolutions constituting 
heat: The author has already applied this hypothesis to the theory of the elasticity 
of gases and vapours (Tratis. Roy. Soc. Edin. vol. xx. part 1). Applied to solids it 
leads to the following conclusion :— 
Theorem Sixth.—In an imperfect solid, each of the coefficients of longitudinal and 
lateral elasticity is equal to the same function of the coefficients of rigidity which 
would have been its value in a perfect solid, added to a coefficient of fluid elasticity, 
which is the same in all directions. 
Thus the number of independent coefficients for such substances is four. The 
rest of the paper is occupied by the deduction from those principles of some im- 
portant consequences respecting coefficients of compressibility and extensibility, and 
the elasticities corresponding to directions not coinciding with the three axes. 
Lieut, Heat, ELectriciry, MAGNETISM. ’ 
On a Method for computing Magnetic Charts of Declination. 
By Samuev Beswick. 
I here exhibit a declination chart for the whole Atlantic for the epoch of 1840, the 
lines of which have been computed by a theoretical formula which I will now de- 
Scribe. It is founded on a principle proposed by Prof. Gauss, which resolves the 
magnetic force of the earth into two portions, one of which, X, acts in the direction 
ef the geographical meridian, and the other, Y, perpendicularly to that meridian. 
Now by a well-known law of the composition of forces, a, resultant force is found 
from the combined action of the above two portions of the horizontal force. But as 
all places of observation must be situated between the two points of convergence of 
the horizontal force, it is clearly necessary that two such equivalent forces must be 
found—one for each hemisphere. These forces acting upon the needle at the place 
of observation will produce an effect proportionate to their comparative angles of 
distaiice ; which effect will be the declination of the needle for that place. Hence 
arises the following formula :— 
e+d Xa. 
a+b 4 
e+y = 2 the declination. 
By this formula I have computed tables of declination, from which has been formed 
lie large niagnetic chart for the whole Atlantic, which,I here exhibit. It presents 
some striking features of comparison with a magnetic chart formed by Col, Sabine 
from tables of observation. This chart is also for the whole Atlantic, and for the 
Same epoch, 1840. And the tables computed by this formula; compared with Sabine’s 
tables of observation, are so similar, that the differences are always within the range 
of errors of observation. 
The process for adapting this formula to all epdchs—which is the grand deside- 
ratum in this department of sciencé—-has been given in detail in the Phil. Mag. for 
March of the present year. The superiority of this method consists in its simplicity, 
precision, and the exactitude of its results. It is capable of computing magnetic 
charts for any epoch. By its means I traced the anomalous Asiatic line of no 
declination for 1800, 1820, 1830, 1835, 1840, 1845, 1849, and 1850; and in each 
case the results give but slight differences from observation. From the successive 
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