Example 2. d2y/dt2 = c2 d2y/dx2. Try y = elx + mt. We find m2 = c2l2, or m = ±cl. Hence y = Ael(x + ct) + Bel(x - ct) is a solution for all values of A, B, l; so, also, is the sum of any number of terms of similar forms. We may infer that the general solution is
y = f(x + ct) + F(x - ct),
where f and F are arbitrary functions. It is only in exceptional cases that an equation can be solved, as in these two examples, by an analytical formula; indeed, differential equations are the most fertile source of new functions in analysis. But, as in the analogous cases of algebraic equations and definite integrals, it may be quite possible to find, by methods of approximation, an arithmetical solution which is sufficient for the purpose in hand.—Bibliography: H. T. H. Piaggio, Differential Equations; J. M. Page, Ordinary Differential Equations; A. R. Forsyth, A Treatise on Differential Equations; E. T. Whittaker and G. N. Watson, Modern Analysis.
Diffrac´tion, a term applied to the bending that rays of light undergo in passing close to the edge of an opaque body. Thus when a beam of direct sunlight is admitted into a dark room through a narrow slit, and falls upon a screen
placed to receive it, there appears a line of white light bordered by coloured fringes; these fringes are produced by diffraction, and in the case given it may be seen that the red or long-wave rays are diffracted more than the blue rays. See Interference.
Diffu´sion, the gradual mixing of gases, liquids, or solids when brought into direct contact. When a block of lead is placed on a block of gold, with their smooth surfaces in close contact, it is found that, after several weeks, gold has diffused into the lead, and lead into the gold. In the case of gases, when a jar of oxygen and a jar of hydrogen are connected together by a tube or opening of any kind, they rapidly become mixed; and their mixture does not depend on gravity, but takes place in opposition to that force, as may be shown by placing the jar of hydrogen gas above the other. Oxygen is sixteen times heavier than hydrogen, bulk for bulk, but the heavier gas moves upwards and the lighter downwards, and the process of intermixture, or diffusion, goes on till the two gases are apparently equably distributed throughout the whole space. After that they have no tendency whatever to separate. Similarly, if two vessels, one containing oxygen and the other hydrogen, be connected by a tube which is stuffed with a plug of porous material, such as plaster of Paris, the gases gradually diffuse one into the other through the porous plug. The two gases, however, do not pass through the porous separator at equal rates, but in inverse proportion to the square roots of the densities of the gases. Thus in the case of two vessels, one containing hydrogen and the other oxygen, which is sixteen times as heavy as hydrogen, the hydrogen will pass towards the oxygen jar four times as quickly as the oxygen will pass towards the hydrogen jar. Kindred phenomena occur when two liquids that are capable of mixing, such as alcohol and water, are put in contact, the two gradually diffusing one into the other in spite of the action of gravity. In some cases, however, as where ether and water are employed, the diffusion is only partial, this result arising from the fact that these two liquids are not miscible in all proportions. When solutions of various solid bodies are placed in contact, interdiffusion also takes place. On the results of his examination of the phenomena of diffusion of liquids and salts across porous membranes or septa, Graham founded a method of separating colloid from crystalloid bodies, which he called dialysis.
Digam´ma, a letter which once belonged to the Greek alphabet, and which remained longest in use among the Æolians. It resembled our letter F, and hence was called digamma, that is, double Γ. It appears to have had the force of f or v. Its existence was first pointed out by Richard Bentley.
Digby, Sir Everard, an English gentleman, born of a Roman Catholic family in 1578. He enjoyed some consideration at the court of Elizabeth and James I, by whom he was knighted. Having contributed money to the Guy Fawkes conspiracy, he was tried and hanged in 1606.
Digby, Sir Kenelm, eldest son of the preceding, born in 1603, died in 1665. He studied at Oxford, was knighted in 1623, and on the accession of Charles I was created a gentleman of the bedchamber, a Commissioner of the Navy, and a governor of Trinity House. He soon after fitted out at his own expense a small but successful squadron against the French and Venetians. In 1636 he became a Roman Catholic, and was imprisoned as a Royalist during 1642-3, when he was allowed to retire to the Continent. At the Restoration he returned to England, became a member of the Royal Society, and was much visited by men of science. He wrote numerous works: a Treatise on the Nature of Bodies, a Treatise on the Nature and Operation of the Soul, and Of the Cure of Wounds by the Powder of Sympathy.
Di´gest, a name originally given to a collection or body of Roman laws, digested or arranged under proper titles by order of the Emperor Justinian. Hence applied to any somewhat similar collection.