Hence the singular equation obtained by the elimination of c between the equations y = f (x, c), dy = 0, represents the locus of such points of successive intersection. In stricter language, the singular solution represents the locus of those points which constitute the limits of position of the points of actual intersection of the different members of the family of curves represented by the equation y =ƒ (x, c), always excepting the case in which that locus coincides with a particular curve of the system. dy And as at these limiting points the value of is the same dx for the locus of the singular solution and the loci of primitives, it follows that the former has contact with every curve of the latter system which it meets. The locus of the singular solution is seen to be the envelope of the loci of primitives. The envelope of the loci of primitives is the locus of a singular solution, except when it coincides with one of the particular loci, of which it forms the connecting bond. Similar observations may be made with reference to the condition = = 0. dx Derivation of the singular solution from the differential equation. 7. We have found that the singular solution of a differential equation considered as derived from its complete primitive possesses the following characters. dy dx 0. dc 1st. It satisfies one of the conditions 0, 2nd. It is not possible to deduce it from the complete. primitive by giving to c a constant value. It has also been shewn that the positive conditions are equivalent except when the singular solution involves only one of the variables in its expression. Now we shall endeavour to translate the above characters from a language whose elements are x, y, and c to a language dy whose elements are x, y, and -from the language of the dx complete primitive to the language of the differential equation. If we differentiate with respect to x the complete primitive expressed in the form and substituting in this for c its expression in terms of x and y given by the primitive (1), we have finally the differential equation in the form Thus the differential equation (3) is the same as the derived equation (2), provided that c be considered therein as a function of x and y determined by (1). provided that the value of the first member be derived from the differential equation, that of the second member from the complete primitive. In like manner if we suppose the complete primitive expressed in the form the first member referring to the differential equation, the second to the complete primitive, The equations (4) and (5), which are rigorous and fundamental, establish a connexion between the differential equation and the complete primitive, and it now only remains to introduce the conditions dy =0. We begin with the former. dc = dx We have seen that when dy = 0 leads to a singular solu dc tion it does so by enabling us to determine c as a function of x, suppose c = X. Before proceeding to more general considerations it will be instructive to make a particular hypothesis as to the form of the equation dy dc = 0. Suppose then this equation to be of the form Q (c-X)=0..... .(6), m being a positive constant and Q a function of x and c, which neither vanishes nor becomes infinite when c = X. This hypothesis is at least sufficiently general to include all the cases in dy which =0 is algebraic, dc and the second term of the right-hand member having c- X for its denominator and not containing c at all in its numerator, is infinite. At the same time, we see that no such infinite term would present itself were c determined as a hand member of (7) being now reduced to its first term. dp The conclusion to which this points is that is infinite for dy a singular solution, but finite for a particular integral. Again, suppose the value of c in terms of x and y furnished by algebraic solution of the complete primitive to be c=4(x, y), then substituting this value in the equation c – X=0, we obtain the singular solution in the form (x, y) - X=0. Now the same substitution gives to the infinite term in the value of dp the form dy We see then, in the case of a singular solution correspond dp ing to a determination c = X, that as derived from the dy differential equation becomes infinite owing to (x, y) − X occurring in a denominator. And, whatever modification of form may be made by clearing of fractions or radicals, we may still infer that, if u =0 be a singular solution derived from an algebraic primitive, the function dp will become infinite, owing dy to u presenting itself under a negative index. The analysis does not however warrant the conclusion that dp any relation between x and y which makes infinite will dy be a solution. If m be a negative constant, the second term in the expression of dp is still infinite, but the prior condition dy dy dc dp dy = 0 is no longer satisfied. All we can affirm is that if = ∞ gives a solution at all it will be a singular solution. dx 1 Since = dy p it is evident that a singular solution originating in a determination of c in the form c=Y_will_make d infinite. dx p conditions =∞, dp d = dx P ∞, is also developed. The former lead to solutions, but not necessarily to singular solutions; the latter do not necessarily lead to solutions, but when they do, those solutions are singular. ential equation. It is therefore a singular solution. It may be objected against the above reasoning, not only that it involves an assumption as to the form of the equa |