results," forming part of Arts. 3 and 4 of Chapter VIII., seem obscure and difficult. The following may perhaps be substituted with advantage. implies that the Thus there can are found from x= constant. singular solution is of the form x = constant. dc = Similarly, if the complete primitive be expressed in the form xy, c), there can be no singular solutions except dF (y, c) In Art. 8 of Chapter VIII. we read, "We may pass over the case in which the above equation is satisfied independently of c, because the relation obtained would involve x only, while it is a condition accompanying the use of dp dy that it leads to solutions involving y at least." It is objected, Why may we pass over this case? Such a case might occur and furnish a solution, and then we should want to know the character of that solution. Take for example dp p=x"y; here if n is negative, dy is infinite when x = 0, and xn+1 this is a singular solution. For the general solution is y=cen+1, and so x=0 is not a case of it. The words-while it is a condition...at least-seem very difficult, for by supposition we are now investigating what is furnished by dp dy =∞ Professor Boole met the objection in substance thus: "It will be found that the rules in the book are correct in this case. What is implied in the Chapter, though not stated with sufficient clearness, is that if do leads to a solution dp dy which does not involve y in its expression, nothing is to be inferred whether it is singular or not. Then the proper test is d Hence x = 0, provided n is between 0 and 1, or y = 0. Consider these separately: First. Let n be between 0 and 1, and x = 0. This is by the test a singular solution. Substituting it in the complete primitive we get y = c, which confirms this. Second. Let y = 0. This satisfies the differential equa tion; but from the fact that it comes from d dx = ∞ we have no inference; from the fact that it does not come from dp d = ∞ we have the inference that it is a particular integral: it dy corresponds to c=0. There remains the case of x=0 when n is between ando. As this does not satisfy d dx () = ∞, we infer that it is a particular integral. To prove this we have When 0 this gives, since 1+ n is negative, x= according as y is positive or negative. This is like Ex. 2 of Chap. VIII. Art. 8." The remark made by Professor Boole in the above reply, that if dp dy = leads to a solution which does not involve y nothing is to be inferred...is important. It corrects the statement put too strongly in Chap. VIII. Art. 7, " All we can affirm dp dy 66 is that if gives a solution at all it will be a singular solution." From Art. 8 onwards it seems assumed that a solution for which dy dc = 0 is always to count as a singular solution, even if it should coincide with a particular integral. This does not seem to have been quite the view of the former part of Chapter VIII. see Arts. 5 and 6 of the Chapter. In Ex. 3 of Art. 9 we read, "the second is obviously a singular solution." This means that since we have a solution which makes infinite, we conclude that it is a singular solution. dp dy So in Ex. 5 of Art. 11 we read, "is evidently "is evidently a singular solution," when it seems better to say, singular solution." "and is therefore a 4. The additional matter relating to Chapter VIII. begins with another example which was to be placed at the close of Art. 3 of that Chapter.] This therefore is the singular solution and it satisfies both the tests, as both x and y are contained in its expression. the first is not satisfied, the last two are satisfied. The determination of c as a function of x by the solution of the equation df (a, c) = 0 is equivalent to determining dc what particular primitive has contact with the envelope at that point of the latter which corresponds to a given value of x. One important remark yet remains. The elimination of c between a primitive y=f(x, c) and the derived equation dy =0, does not necessarily lead to a singular solution in the dc sense above explained. For it is possible that the derived equation may neither on the one hand enable us to determine c as a function of x, so leading to a singular solution; nor, on the other hand, as an absolute constant, so leading to a particular primitive. Thus the particular primitive whence c is if x be negative, and if x be positive. It is a dependent constant. The resulting solution y = 0 does not then represent an envelope of the curves of particular primitives, nor strictly one of those curves. It represents a curve formed of branches from two of them. It is most fitly characterized as a particular primitive marked by a singularity in the mode of its derivation from the complete primitive. All the foregoing observations and conclusions may be extended to the case of solutions derived from the condition by an absolutely constant value of c, so leading to a particular primitive and not a singular solution. In this case (x+h, c) as well as (a, c) would vanish, and the numerator of (9), instead of being the difference of a finite and an infinite quantity, would be the difference of two infinite and equal quantities. [See Chap. VIII. Art. 8.] It would not there fore be infinite. Hence we conclude that would not become dp dy infinite for a particular primitive in the strict sense of that |