dP squaredS dP squaredS
(r,s) = ----------, (r,s') = -----------, (3)
dPq_rdPq_s dPq_rdPq'_s
we have
dPp_r = - [Sigma]_s(r,s)[delta]q_s - [Sigma]_s(r,s')[delta]q'_s (4)
with a similar expression for [Delta]p_r. Hence the right-hand side of
(2) becomes
- [Sigma]_r{[Sigma]_s(r,s)[delta]q_s + [Sigma]_s(r,s')[delta]q'_s}[Delta]q_r
+ [Sigma]_r{[Sigma]_s(r,s)[Delta]q_s + [Sigma]_s(r,s')[Delta]q'_s}[delta]q_r
= [Sigma]_r[Sigma]_s(r,s'){[delta]q_r.[Delta]q'_s - [Delta]q_r.[delta]q'_s}. (5)
The same value is obtained in like manner for the expression on the
left hand of (2); hence the theorem, which, in the form (1), is due to
Lagrange, and was employed by him as the basis of his method of
treating the dynamical theory of _Variation of Arbitrary Constants_.
Helmholtz's reciprocal theorems.
The formula (2) leads at once to some remarkable reciprocal relations
which were first expressed, in their complete form, by Helmholtz.
Consider any natural motion of a conservative system between two
configurations O and O' through which it passes at times t and t'
respectively, and let t' - t = [tau]. As the system is passing through
O let a small impulse [delta]p_r be given to it, and let the
consequent alteration in the co-ordinate q_s after the time [tau] be
[delta]q'_s. Next consider the _reversed_ motion of the system, in
which it would, if undisturbed, pass from O' to O in the same time
[tau]. Let a small impulse [delta]p'_s be applied as the system is
passing through O', and let the consequent change in the co-ordinate
q_r after a time [tau] be [delta]q_r. Helmholtz's first theorem is to
the effect that
[delta]q_r : [delta]p'_s = [delta]q'_s : [delta]p_r. (6)
To prove this, suppose, in (2), that all the [delta]q vanish, and
likewise all the [delta]p with the exception of [delta]p_r. Further,
suppose all the [Delta]q' to vanish, and likewise all the [Delta]p'
except [Delta]p'_s, the formula then gives
[delta]p_r.[Delta]q_r = - [Delta]p'_s.[delta]q'_s, (7)
which is equivalent to Helmholtz's result, since we may suppose the
symbol [Delta] to refer to the reversed motion, provided we change
the signs of the [Delta]p. In the most general motion of a top
(MECHANICS, Sec. 22), suppose that a small impulsive couple about the
vertical produces after a time [tau] a change [delta][theta] in the
inclination of the axis, the theorem asserts that in the reversed
motion an equal impulsive couple in the plane of [theta] will produce
after a time [tau] a change [delta][psi], in the azimuth of the axis,
which is equal to [delta][theta]. It is understood, of course, that
the couples have no components (in the generalized sense) except of
the types indicated; for instance, they may consist in each case of a
force applied to the top at a point of the axis, and of the
accompanying reaction at the pivot. Again, in the corpuscular theory
of light let O, O' be any two points on the axis of a symmetrical
optical combination, and let V, V' be the corresponding velocities of
light. At O let a small impulse be applied perpendicular to the axis
so as to produce an angular deflection [delta][theta], and let ss'
be the corresponding lateral deviation at O'. In like manner in the
reversed motion, let a small deflection [delta][theta]' at O' produce
a lateral deviation ss at O. The theorem (6) asserts that
ss ss'
----------------- = ---------------, (8)
V'[delta][theta]' V[delta][theta]
or, in optical language, the "apparent distance" of O from O' is to
that of O' from O in the ratio of the refractive indices at O' and O
respectively.
Helmholtz's second reciprocal theorem.
In the second reciprocal theorem of Helmholtz the configuration O is
slightly varied by a change [delta]q_r in one of the co-ordinates, the
momenta being all unaltered, and [delta]q'_s is the consequent
variation in one of the momenta after time [tau]. Similarly in the
reversed motion a change [delta]p'_s produces after time [tau] a
change of momentum [delta]p_r. The theorem asserts that
[delta]p'_s : [delta]q_r = [delta]p_r : [delta]q'_s (9)
This follows at once from (2) if we imagine all the [delta]p to
vanish, and likewise all the [delta]q save [delta]q_r, and if
(further) we imagine all the [Delta]p' to vanish, and all the
[Delta]q' save [Delta]q'_s. Reverting to the optical illustration, if
F, F', be principal foci, we can infer that the convergence at F' of a
parallel beam from F is to the convergence at F of a parallel beam
from F' in the inverse ratio of the refractive indices at F' and F.
This is equivalent to Gauss's relation between the two principal focal
lengths of an optical instrument. It may be obtained otherwise as a
particular case of (8).
We have by no means exhausted the inferences to be drawn from
Lagrange's formula. It may be noted that (6) includes as particular
cases various important reciprocal relations in optics and acoustics
formulated by R.J.E. Clausius, Helmholtz, Thomson (Lord Kelvin) and
Tait, and Lord Rayleigh. In applying the theorem care must be taken
that in the reversed motion the reversal is complete, and extends to
every velocity in the system; in particular, in a cyclic system the
cyclic motions must be imagined to be reversed with the rest.
Conspicuous instances of the failure of the theorem through incomplete
reversal are afforded by the propagation of sound in a wind and the
propagation of light in a magnetic medium.
It may be worth while to point out, however, that there is no such
limitation to the use of Lagrange's formula (1). In applying it to
cyclic systems, it is convenient to introduce conditions already laid
down, viz. that the co-ordinates q_r are the palpable co-ordinates and
that the cyclic momenta are invariable. Special inference can then be
drawn as before, but the interpretation cannot be expressed so neatly
owing to the non-reversibility of the motion.
AUTHORITIES.--The most important and most accessible early authorities
are J.L. Lagrange, _Mecanique analytique_ (1st ed. Paris, 1788, 2nd
ed. Paris, 1811; reprinted in _Oeuvres_, vols. xi., xii., Paris,
1888-89); Hamilton, "On a General Method in Dynamics," _Phil. Trans._
1834 and 1835; C.G.J. Jacobi, _Vorlesungen ueber Dynamik_ (Berlin,
1866, reprinted in _Werke_, Supp.-Bd., Berlin, 1884). An account of
the extensive literature on the differential equations of dynamics and
on the theory of variation of parameters is given by A. Cayley,
"Report on Theoretical Dynamics," _Brit. Assn. Rep._ (1857),
_Mathematical Papers_, vol. iii. (Cambridge, 1890). For the modern
developments reference may be made to Thomson and Tait, _Natural
Philosophy_ (1st ed. Oxford, 1867, 2nd ed. Cambridge, 1879); Lord
Rayleigh, _Theory of Sound_, vol. i. (1st ed. London, 1877; 2nd ed.
London, 1894); E.J. Routh, _Stability of Motion_ (London, 1877), and
_Rigid Dynamics_ (4th ed. London, 1884); H. Helmholtz, "Ueber die
physikalische Bedeutung des Prinzips der kleinsten Action," _Crelle_,
vol. c., 1886, reprinted (with other cognate papers) in _Wiss. Abh._
vol. iii. (Leipzig, 1895); J. Larmor, "On Least Action," _Proc. Lond.
Math. Soc._ vol. xv. (1884); E.T. Whittaker, _Analytical Dynamics_
(Cambridge, 1904). As to the question of stability, reference may be
made to H. Poincare, "Sur l'equilibre d'une masse fluide animee d'un
mouvement de rotation" _Acta math._ vol. vii. (1885); F. Klein and A.
Sommerfeld, _Theorie des Kreisels_, pts. 1, 2 (Leipzig, 1897-1898); A.
Lioupanoff and J. Hadamard, _Liouville_, 5me serie, vol. iii. (1897);
T.J.I. Bromwich, Proc. Lond. Math. Soc. vol. xxxiii. (1901). A
remarkable interpretation of various dynamical principles is given by
H. Hertz in his posthumous work _Die Prinzipien der Mechanik_
(Leipzig, 1894), of which an English translation appeared in 1900.
(H. Lb.)
DYNAMITE (Gr. [Greek: dynamis], power), the name given to several
explosive preparations containing nitroglycerin (q.v.) which are almost
exclusively used for blasting purposes. The first practical application
of nitroglycerin in this way was made by A. Nobel in 1863. He soaked
gunpowder with the liquid and fired the gunpowder by an ordinary fuse.
Later he found that nitroglycerin could be detonated by the explosion of
several materials such as fulminate of mercury, the use of which as a
detonator he patented in 1867. In 1866-1867 he experimented with
charcoal and other substances, and found the infusorial earth known as
kieselguhr, which consists mainly of silica (nearly 95%), eminently
adapted to the purpose, as it was inert, non-combustible, and after a
little heating and preparation very porous, retaining a large amount of
nitroglycerin as water is held in a sponge, without very serious
exudation on standing. This kieselguhr dynamite is generally made by
incorporating three parts of nitroglycerin with one part of the dry
earth, the paste being then formed into cylindrical cartridges. This
work is done by hand. Generally a small percentage of the kieselguhr is
replaced by a mixture containing sodium and ammonium carbonates, talc
and ochre. This product is known as dynamite No. 1. Disabilities
attaching to kieselguhr dynamite are that when placed in water the
nitroglycerin is liable to be exuded or displaced, also that, like
nitroglycerin itself, it freezes fairly easily and thawing the frozen
cartridges is a dangerous operation. Other substances, e.g. kaolin,
tripoli, magnesia alba (magnesium carbonate), alumina, sugar, charcoal,
some powdered salts and mixtures of sawdust and salts, have been shown
to be absorbents more or less adapted to the purpose of making a
dynamite. Charcoal from cork is said to absorb about 90% of its weight
of nitroglycerin. With the idea of obtaining greater safety, mixtures
have been made of nitroglycerin with wood fibre, charcoal and metallic
nitrates. Lithofracteur, for instance, consists of 50% nitroglycerin and
a mixture of prepared sawdust, kieselguhr and barium nitrate. Carbonite
contains 25% of nitroglycerin, the remainder being a mixture of
wood-meal and alkali nitrates, with about 1% of sulphur. Dualin, atlas
dynamite and potentite are other modifications.
A convenient form in which nitroglycerin can be made up for blasting
purposes, especially in wet ground, is the gelatinous material obtained
by the action of nitroglycerin, either alone or with the help of
solvents, on low-grade or soluble gun-cottons. It is known as blasting
gelatin, and was first made by Nobel by incorporating 6 or 7% of low
nitrated cellulose (collodion cotton or soluble gun-cotton) with
slightly warmed nitroglycerin. The result is a transparent plastic
material, of specific gravity 1.5 to 1.6, which may be kept under water
for a long time without appreciable change. It is less sensitive to
detonation than ordinary dynamite, and although its explosion is
slightly slower it is more powerful than dynamite and much superior to
the liquid nitroglycerin. Blasting gelatin also freezes and is sensitive
to percussion in this state. Camphor and other substances have been
added to blasting gelatin to render it more solid and less sensitive.
Some modifications of blasting gelatin, e.g. gelignite, contain
wood-meal and such oxygen-containing salts as potassium nitrate.
Experience has conclusively shown that dynamites are more satisfactory,
quicker, and more intense in action than liquid nitroglycerin.
To prevent nitroglycerin and some of the forms of dynamite from freezing
it has been proposed to add to them small quantities of either
monochlor-dinitroglycerin or of a nitrated poly-glycerin. The former is
obtained by first acting upon glycerin with hydrogen chloride to produce
_u-_chlorhydrin or chlor-propylene glycol, C3H7O2Cl, which is then
nitrated as in the case of glycerin. The latter is obtained by heating
glycerin for six or seven hours to about 300 deg. C., whereby water is split
off in such manner that a diglycerin C6H14O5, for the most part,
results. This on nitration in the usual manner gives a product
C6H{10}N4O{13}, which burns and explodes in a similar manner to ordinary
nitroglycerin, but is less sensitive and does not so easily freeze. The
mono- and di-nitrates of glycerin have also been proposed as additions
to ordinary nitroglycerin (q.v.) for the same purpose. (W. R. E. H.)
DYNAMO (a shortened form of "dynamo-electric machine," from Gr. [Greek:
dynamis], power), a machine for converting mechanical into electrical
energy.
The dynamo ranks with the telegraph and telephone as one of the three
striking applications of electrical and magnetic science to which the
material progress that marked the second half of the 19th century was in
no small measure due. Since the discovery of the principle of the dynamo
by Faraday in 1831 the simple model which he first constructed has been
gradually developed into the machines of 5000 horse-power or more which
are now built to meet the needs of large cities for electric lighting
and power, while at the same time the numbers of dynamos in use have
increased almost beyond estimate. Yet such was the insight of Faraday
into the fundamental nature of the dynamo that the theory of its action
which he laid down has remained essentially unchanged. His experiments
on the current which was set up in a coil of wire during its movement
across the poles of a magnet led naturally to the explanation of induced
electromotive force as caused by the linking or unlinking of magnetic
lines of flux with an electric circuit. For the more definite case of
the dynamo, however, we may, with Faraday, make the transition from
line-linkage to the equivalent conception of "line-cutting" as the
source of E.M.F.--in other words, to the idea of electric conductors
"cutting" or intersecting[1] the lines of flux in virtue of relative
motion of the magnetic field and electric circuit. On the 28th of
October 1831 Faraday mounted a copper disk so that it could be rotated
edgewise between the poles of a permanent horse-shoe magnet. When so
rotated, it cut the lines of flux which passed transversely through its
lower half, and by means of two rubbing contacts, one on its periphery
and the other on its spindle, the circuit was closed through a
galvanometer, which indicated the passage of a continuous current so
long as the disk was rotated (fig. 1). Thus by the invention of the
first dynamo Faraday proved his idea that the E.M.F. induced through the
interaction of a magnetic field and an electric circuit was due to the
passage of a portion of the electric circuit _across_ the lines of flux,
or vice versa, and so could be maintained if the cutting of the lines
were made continuous.[2] In comparison with Faraday's results, the
subsequent advance is to be regarded as a progressive perfecting of the
mechanical and electro-magnetic design, partly from the theoretical and
partly from the practical side, rather than as modifying or adding to
the idea which was originally present in his mind, and of which he
already saw the possibilities.
[Illustration: FIG. 1.]
A dynamo, then, is a machine in which, by means of continuous relative
motion, an electrical conductor or system of conductors forming part of
a circuit is caused to cut the lines of a magnetic field or fields; the
cutting of the magnetic flux induces an electromotive force in the
conductors, and when the circuit is closed a current flows, whereby
mechanical energy is converted into electrical energy.
Little practical use could be made of electrical energy so long as its
only known sources were frictional machines and voltaic batteries. The
cost of the materials for producing electrical currents on a large
scale by chemical action was prohibitive, while the frictional machine
only yielded very small currents at extremely high potentials. In the
dynamo, on the other hand, electrical energy in a convenient form
could be cheaply and easily obtained by mechanical means, and with its
invention the application of electricity to a wide range of commercial
purposes became economically possible. As a converter of energy from
one form to another it is only surpassed in efficiency by another
electrical appliance, namely, the transformer (see TRANSFORMERS). In
this there is merely conversion of electrical energy at a high
potential into electrical energy at a low potential, or vice versa,
but in the dynamo the mechanical energy which must be applied to
maintain the relative movement of magnetic field and conductor is
absorbed, and reappears in an electrical form. A true transformation
takes place, and the proportion which the rate of delivery of
electrical energy bears to the power absorbed, or in other words the
_efficiency_, is the more remarkable. The useful return or "output" at
the terminals of a large machine may amount to as much as 95% of the
mechanical energy which forms the "input." Since it needs some prime
mover to drive it, the dynamo has not made any direct addition to our
sources of energy, and does not therefore rank with the primary
battery or oil-engine, or even the steam-engine, all of which draw
their energy more immediately from nature. Yet by the aid of the
dynamo the power to be derived from waterfalls can be economically and
conveniently converted into an electrical form and brought to the
neighbouring factory or distant town, to be there reconverted by
motors into mechanical power. Over any but very short distances energy
is most easily transmitted when it is in an electrical form, and
turbine-driven dynamos are very largely and successfully employed for
such transmission. Thus by conducing to the utilization of water-power
which may previously have had but little value owing to its
disadvantageous situation, the dynamo may almost be said to have added
another to our available natural resources.
The two essential parts of the dynamo, as required by its definition,
may be illustrated by the original disk machine of Faraday. They are (1)
the _iron magnet_, between the poles of which a magnetic field exists,
and (2) the _electrical conductors_, represented by the rotating copper
disk. The sector of the disk cutting the lines of the field forms part
of a closed electric circuit, and has an E.M.F. induced in it, by reason
of which it is no longer simply a conductor, but has become "active." In
its more highly developed form the simple copper disk is elaborated into
a system of many active wires or bars which form the "winding," and
which are so interconnected as to add up their several E.M.F.'s. Since
these active wires are usually mounted on an iron structure, which may
be likened to the keeper or "armature" of a magnet rotating between its
poles, the term "armature" has been extended to cover not only the iron
core, but also the wires on it, and when there is no iron core it is
even applied to the copper conductors themselves. In the dynamo of
Faraday the "armature" was the rotating portion, and such is the case
with modern continuous-current dynamos; in alternators, however, the
magnet, or a portion of it, is more commonly rotated while the armature
is stationary. It is in fact immaterial to the action whether the one or
the other is moved, or both, so long as their relative motion causes the
armature conductors to cut the magnetic flux. As to the ultimate reason
why an E.M.F. should be thereby induced, physical science cannot as yet
yield any surer knowledge than in the days of Faraday.[3] For the
engineer, it suffices to know that the E.M.F. of the dynamo is due to
the cutting of the magnetic flux by the active wires, and, further, is
proportional to the rate at which the lines are cut.[4]
[Illustration: FIG. 2.]
The equation of the _electromotive force_ which is required in order to
render this statement quantitative must contain three factors, namely,
the density of the flux in the air-gap through which the armature
conductors move, the active length of these wires, and the speed of
their movement. For given values of the first and third factors and a
single straight wire moved parallel to itself through a uniform field,
the maximum rate of cutting is evidently obtained when the three
directions of the lines of the conductor's length and of the relative
motion are respectively at right angles to each other, as shown by the
three co-ordinate axes of fig. 2. The E.M.F. of the single wire is then