CHAPTER XXIII. ADDITIONS TO CHAPTER X. 1. THE theory of singular solutions of differential equations of the higher orders has been presented in the most complete form which it has yet received by Legendre. (Mémoires de l'Académie Royale des Sciences, 1790, p. 218.) He determines first the possible forms of these solutions considered as emerging from the complete primitive by the variations of its arbitrary constants, and secondly the theory of their derivation from the differential equation itself. I shall follow the same order, and shall in the end endeavour to point out in what respect Legendre's theory may be regarded as complete, and in what respect it is imperfect. Suppose the differential equation to be of the nth th order, and let it when solved with respect to the highest differential coefficient of y be represented by Let also its complete primitive, solved with respect to y, be represented by y = f(x, α1, α2 an) ... (2), a, a,... a, being the arbitrary constants of the solution. If we differentiate (2) with respect to x, regarding a, a,,... an no longer as constants but as functions of x, so to be determined as to leave the expressions for y1, Y2,... Yn as functions of a1, a2 an the same as before, we shall have, on representing the second member of (2) by f, df da, df da, df dan + + = 0. da, dx da dx dan dx Differentiating on the same hypothesis the first of these two equations, we find in the same way will be satisfied, i.e., that y1, y,... Yn will have the same expressions when a, a,,... a, are variable as they have when these are constant, provided that the law of their variation be determined by the conditions are known functions of x, a, a,,... a, when the form of ƒ is known. we have a relation between x, a, a,... an; and this relation, with the given complete primitive and the first n-1 of the derived and reduced equations, viz., with will enable us to eliminate a, a,,... a,, and to obtain a relation of the form This is a differential equation of the (n - 1)th order. It differs in its origin from the given differential equation, in that a new relation between x, a, a,,... a, has been employed in place of the nth equation, derived by differentiation from the complete primitive, for the elimination of the constants. The differential equation of the (n-1)th order thus obtained has an integral expressing y in terms of x, and n- 1 arbitrary constants. This is the most general form of a singular solution of the differential equation. - It is possible that the elimination of a, a,... a, may lead to a resulting differential equation which, instead of being of the order n 1, is of the order n - 2, n − 3, &c. The complete integral of such equation would be a singular solution of the differential equation. These possible types of solutions are distinguished by Legendre according to the number of arbitrary constants which they contain. A solu tion containing n-1 arbitrary constants is called by him a singular solution of the first order; one containing n - 2 arbitrary constants a singular solution of the second order, and so on. Adopting this language we might term the complete primitive a singular solution of the order 0. Lastly, any relation between x and y, which satisfies the given differential equation, will constitute a particular case, either of the complete primitive or of one of the general forms of singular solutions above defined. In the case of differential equations of the first order it is seen that no arbitrary constant can appear in the expression of the singular solution. ax2 y= + bx+a2 + b2 required its singular solution. Proceeding as above, we find on the hypothesis of a and b being variable parameters, the same formal expressions for dy d'y as if those parameters were constant, viz. provided that the variation of a and b be such as to satisfy |