1817.) Dilatation of Liquids at all Temperatures. 445 
mulas include no hypothesis, and are deduced solely from observa- 
tions, it seems better to adhere to them, and to refer the dilatation 
of other liquids, as we have done, to that of mercury, as including 
the first three powers of the temperature which the thermometer 
indicates. 
M. Laplace, whose views in physics are always so ingenious and 
general, has engaged me to examine whether it would not be pos- 
sible to expunge the term depending on the cube of the tempera- 
ture, by referring all the dilatations to an ideal thermometer, such 
that the mercurial thermometer itself should be expressed in a 
function of it in the same manner bya simple law of squares, 
reckoning for each liquid from a different point. But I have ascer- 
tained that this agreement is not general, at least with the coeffi- 
cients which I have obtained ; for their signs change for the diffe- 
rent liquids as well as their values, so that it would be impossible to 
make the term depending on the cube of the temperature to disap- 
pear in all these liquids, by any single supposition respecting the 
dilatation of mercury in a function of an ideal thermometer. Perhaps 
experiments still more exact than those which I have used may. 
enable us hereafter to discover a more simple law ; but till that time 
come, the formulas which I have given will supply the occasions of 
observers. 
ArrTIcLe IV. 
On the Calculus of Variations. ‘Translated from Traité de Calcul 
Integral, par Bossut. By Mr. George Harvey, of Plymouth. 
(To Dr. Thomson.) 
SIR, ' 
Tue following exposition of the theory of variations is translated 
from the Calcul Integral of Bossut. I should not request its inser- 
tion in your Annals, did I not conceive that its publication would 
be found eminently beneficial to the young mathematician. 
Iam, Sir, your humble servant, 
Plymouth, April 26, 1817. GrorcEe Harvey. 
—— 
Let there be any function whatever, composed of constant and 
variable quantities, which changes its value either by the zncrease 
or decrease of one or more of the elements which it contains. It 
will therefore undergo a variation; and the method of determining 
this variation is denominated the calculus of variations. 
The variation of a function is designated by the Greek letter 3, in 
the same way as the differential is denoted by the Roman letter d ; 
and the fundamental rules of the calculus of variations rest on the 
same principles as those of the differential calculus; but it is neces- 
