When the instrument is used, the metallic bar to be experimented on is placed in the cavity of the register, and the index pressed down upon it and firmly fixed in its place by the platinum strap and porcelain wedge. The scale is then applied by carefully adjusting the frame to the register and fixing it by pressing a' upon the shoulder. Holding the whole together steadily in the left hand, the lighter arm is so placed that the steel point h may rest upon the edge of the index, against which it will be pressed by the spring; then by slightly turning the larger arm, the point will move along the surface of the index till it drops into the cavity t. The indications of the vernier being then read off, the register is detached from the scale, placed in the furnace, and after it is removed and cooled, it is again applied to the scale in the same Manner as before, and the second indication of the vernier noted. From the two readings of the vernier may be deduced the excess of the expansion of the metallic bar above that of the black. lead, though a correct formula for .this purpose has not, to the writer's knowledge, been hitherto given.
The one employed by Mr. Daniell, though probably sufficiently correct for all practical purposes, gives the expansions one per cent., too great without exception, and in many cases much more, so that more than the first significant figure can seldom be depended upon in those published by him in the Phil. Trans.' of 1830-31. The error thus introduced is perhaps within the limits of the error to which the Instrument itself is liable ; but should this not be the case, it might be desirable to employ the correct formula, for which reason we subjoin Its investigation.
Let c n n, h ny, represent the positions of the two arms of the scale relative to the register, before the expansion has taken place, and h'n'g', their positions after the expansion ; h and le the two positions of the steel point, the line joining which passes through the fixed centre c ; e and e the two positions of the zero of the graduated arc. Put the angle c n 13= c n' re= a ; enn= n' = ; eng (the first reading of the vernier)=4, ; a' WI( (the second reading)= 0'; also c it= c n'=n1t= n'k= r ; and h h' = tho excess of the expansion of the metal above that of the black lead : then It consists of two distinct parts, the register and the scale. The register Is a solid bar of black-lead earthenware, n D D D, eight inches long and seven-tenths of an inch wide and thick, cut out of a common black-lead crucible. In this a hole is drilled three-tenths of an inch in diameter, and seven inches and a half in depth. At pp the upper end of this Isar, and on one of its sides about six-tenths of an inch in length of its substance, are cut away to the depth of half the diameter of the bore. When a bar of any metal six inches and a half long is dropped into this cavity, it rests against its solid end ; and a cylindrical piece of porcelain, q, about one Inch and a half long, called the index, is placed on the top of it, which, projecting into and beyond the open part, is firmly confined to its place by a strap of platinum, r, which passing round the black-lead bar and over the piece of porcelain, is made to press upon the latter with any required degree of tension by means of a small wedge a of porcelain inserted between the bar and the strap. When the register is exposed to the heat of a furnace, it is
evident that, the expansion of the metallic bar exceeding that of the black-lead, the porcelain index will be forced forward ; and when the register is afterwards cooled, the tension of the strap will retain the index at the point of greatest elongation.
The object of the scale is the accurate measurement of the distance through which the index has advanced. It consists of a frame, a a a a, composed of two rectangular plates of brass joined at right angles by their edges, and fitting square upon two aides of the register. At one extremity of this frame is a small plate of brass, a', which, when the two former plates are applied to the register, is brought down upon the shoulder formed by cutting away the black-lead at p, and the whole may be thus firmly adjusted, when required, to the black-lead bar by three planes of contact. To the outside of this frame is firmly attached, by means of the screws bb, a brass plats a the extremity where the unit of measurement is one inch.
The formula used by Mr. Daniell is e= sin 4 (os--4)), or its equiva lent, since is generally a small angle, sin from which it appears that all the expansions given by him should be diminished in the ratio of 1 : cos 7° 30' 4- (˘"+ ; but as he has recorded only the difference 4/-0 of the readings of the vernier, and not the readings themselves, this correction can only be made by a repetition of the whole of the experiments. The error is inconsiderable so long as ˘ and are both small, but it increases with the increase of either of those angles.
The excess of the expansion of the metal above that of the black lead being thus obtained, and increased by the expansion of the latter (the determination of which is less direct and conclusive), the expan sion of the metal becomes known. In order that the instrument may then be employed as a measure of temperature as well as of expansion, the doubtful assumption is introduced that equal increments of length are the effects of equal increments of temperature, and thence, having determined the expansion between any two known points on the thermometric scale, any the temperatures of melting ice and boiling mercury, • mere proportion will of course give the temperature at which any other observed expansion took place.