and the last result is called the solution of the given equation, although it involves an integration which cannot be performed in finite terms. The relation among the variables which constitutes the general solution of a differential equation, as above described, is also termed its complete primitive. The relation (14) involving the arbitrary constant c is virtually the complete primitive of the differential equation (13). It will be observed that the terms 'general solution' and 'complete primitive,' though applied to a common object, have relation to distinct processes and to a distinct order of thought. In the strict application of the former term we contemplate the differential equation as prior in the order of thought, and the explicit relation among the variables as thence deduced by a process of solution; while in the strict use of the latter term the order both of thought and of process is reversed. Genesis of Differential Equations. 6. The theory of the genesis of differential equations from their primitives is to a certain extent explained in treatises on the Differential Calculus, but there are some points of great importance relating to the connexion of differential equations thus derived, not only with their primitive, but with each other, which need a distinct elucidation. Suppose that the complete primitive expresses a relation between x, y and an arbitrary constant c. Differentiating on the supposition that x is the independent variable, we obtain a new equation which must involved, and which may involve any or all of the quantities x, y and c. If it do not involve c, it will constitute the differential equation of the first order corresponding to the given primitive. If it involve c, then the elimination of c between it and the primitive will lead to the differential equation in question. Thus if the complete primitive be the differential equation of the first order of which (1) is the complete primitive. That primitive might have been so prepared as to lead to the same final equation by mere differentiation. Thus, reducing the primitive to the form we have on differentiating and clearing the result of fractions, dy which agrees with (3). And generally, if a primitive involving an arbitrary constant c be reduced to the form (x, y) = c, the corresponding differential equation will be obtained by mere differentiation and removal of irrelevant factors, i. e. of factors which do not contain dy dx the relation in which stands to x and y. For it is in that relation, as already intimated, Art. 2, that the essential character of the differential equation consists. It is to be observed that when the differentiation of a primitive involving an arbitrary constant c does not of itself cause that constant to disappear, the result to which it leads is still a differential equation, only not that differential equation of which the equation given constitutes the complete primitive. Thus, while the complete primitive of (3) is (1), that of (2) is y= cx+c', c' being now the arbitrary constant, arbitrary as being independent of anything contained in the differential dy equation. Indeed when we consider =c as the differential dx equation, the constant c, as entering into its complete primitive, y = cx + c', is not arbitrary, the value which it bears in the primitive being determined by that which it bears in the differential equation. As another illustration of the same theory, the equation y=ce as complete primitive gives rise to the differential equation of the first order while the equation immediately derived from it by differdy entiation, viz, = cae, has for its complete primitive dx y = ce* + c'. To the last mentioned differential equation, y=ce stands in the relation of a particular primitive. ax Second and Higher Orders. 7. It is shewn in the previous section that from an equation containing x and y with an arbitrary constant c, we can by differentiation, and elimination (if necessary) of that constant, obtain the differential equation of the first order, of which the given equation constitutes the complete primitive. In like manner an equation connecting x, y, and two arbitrary constants being given, if we differentiate twice, and eliminate, should they not have already disappeared, the arbitrary constants, we shall arrive at a differential equation of the second order free from both the constants in question, and of which the given equation constitutes the complete primitive. Thus, if we take as the primitive equation a differential equation of the first order free from the constant b. Differentiating this equation we have and, eliminating a between the last two equations, a differential equation of the second order free from both a and b. In the above example the constant b was eliminated after the first differentiation, and the constant a after the second. But the same final result would have been arrived at if the order of the eliminations had been reversed. Thus, if a be eliminated between (4) and (5), we shall have a differential equation of the first order, different in form from (6), and involving b instead of a. But on differentiating this equation and eliminating b, we shall arrive at the same final equation of the second order (7). And generally the order in which the constants are eliminated does not affect the form of the final differential equation. Now a little consideration will shew that this is necessarily the case. We are to remember that the generality which the primitive derives from the presence of its arbitrary constants consists only in this, that it is thus made to stand as the representative of an infinite number of particular equations, in each of which these constants receive particular and definite values. If in any one of the equations thus particularized we further give to a definite value, definite values will also result for y, x' dx2 dy d'y &c. Thus to a given abscissa of a given curve, i.e. of a curve determined as to its species by the form of its equation, and as to its elements by the values of the constants in that equation, correspond only definite values of the. ordinate y determining the corresponding points of the curve, definite values of determining the inclination of the tandx gents at such points to the axis of x, and definite values of determining, in conjunction with the former, the measure of curvature at the same points. In other words, the species of the curve as defined by an equation of the form & (x, y, a, b) = 0 being fixed, the values of y, dx' d dy d'y have a fixed dependence d'y dx dy on those of a, b and x. And hence the equation (x, y, a, b) = 0 being given, any processes of differentiation, elimination, &c. applied thereto can only serve, either 1st, to bring out or manifest the dependence above referred to, or 2ndly, to modify the accidental form of its expression; but in no sense to create such dependence or affect its real nature. Now this dependence of y, de' d upon a, b, dy d'y and x, involves the existence of three equations among six quantities. Therefore the elimination which thus becomes possible of two of those quantities, a, b, must leave a single final relation between the remaining four, x, y, de da・ dy d'y And this is the differential equation in question. As another example, let us eliminate the arbitrary constants c and c from the equation |