Theory of the Second Integration.

9. First suppose the values of m unequal.

Then u1 =α1, v=b, being the two integrals (and we have seen that there cannot be more than two) of one of the systems of linear partial differential equations, and u, = a,, v=b2 those of the other, the general first integrals of the given system will be

(u1, v1) = 0, ¥ (U2, v2) = 0.

The values of P and q determined from these will by Proposition IV. render

dz-pdx - qdy = 0

integrable, and the integral of this will be the general integral of the proposed partial differential equation. For it will involve explicitly or implicitly two arbitrary functions derived from those in the first integrals.

It suffices however, following herein Charpit's method, to combine one general first integral derived from the one system with a particular first integral derived from the other system, e.g. the integrals

The values of p and q hence derived, and employed as before, will lead to a second integral involving one arbitrary function and containing two arbitrary constants. This constitutes a complete primitive from which the general solution will be obtained by converting one of the arbitrary constants into an arbitrary function of the other, and eliminating the latter between the equation and the one derived from it by differentiation with respect to that constant.

Secondly, suppose the values of m equal.

In this case we have but one system of partial differential equations so constituted however that if it admits of two integrals it will admit of three.

Let u= a, v=b, wc represent these integrals. Then if from these we eliminate p and q we shall obtain a final integral of the form

z=ƒ (x, y, a, b, c),

and this constitutes a complete primitive from which we shall deduce the general integral by making b=4(a), c = ¥ (a), and eliminating a between the equations

To prove this let us combine the general and particular first integrals

v = =$(u), u = a.

The values of p and q hence obtained make

dz-pdx - qdy = 0

integrable, and the result can be no other than the remaining integral wc, or rather what this would become on eliminating p and q from it. But since the equations by which this integration are to be effected are equivalent to

u = a, v = $(a),

w will become a function of x, y, z, a and p (a). Also by Charpit's method c is to be treated as a function of a, so that ultimately we have the result above assigned.

We have here supposed U not to vanish. If it do the theory assumes another but simpler form. Let

v=f(u), w = (u)

be the two general first integrals. Then, since by the condition at the close of Art. 2, if p be eliminated from these equations q will also disappear, it suffices to eliminate them together in order to obtain the general second integral.

10. Although the cases in which U=0 and V=0 have in the foregoing sections been treated for simplicity apart, their theory might have been deduced from that of the case in which neither Unor V vanishes.

Thus to deduce the equations for the case of U= 0 elimi

This is equivalent to the results of Art. 5, Case 1.

11. We found it necessary (Art. 3) in order that the general partial differential equation of this Chapter should be satisfied by the envelope of a system of surfaces the equations of which contain three parameters varying under two conditions that the relation

should be satisfied.

S2 + 4 (UV — RT) = 0

It appears from Art. 8 that this is but one of three conditions necessary and together sufficient for this purpose. The formal conditions for every form of ultimate solution consistent with the existence of a general first integral F (u, v) = 0 can be deduced in the same way.

[In the Bulletin de l'Académie Impériale des Sciences de St Pétersbourg, Vol. IV. 1862, there is an article entitled Considérations sur la recherche des integrales premières des équations différentielles partielles du second ordre, par G. Boldt (Lu le 7 Juin 1861).

The article occupies pages 198-215 of the volume. Although the name does not quite correspond, I consider that to be a misprint, and I attribute the article to Professor Boole, partly from the nature of the contents, and partly because it is known by his friends that he was engaged at a time corresponding to the date here given in the preparation of a mathematical article in French.

The object of the article is to determine the conditions necessary for the existence of a first integral of the equation

dz

dz

where R, S, T, and Ware any functions of x, y, z, and ;

dx dy

and also to determine the conditions which must hold in order that Ampère's method of integration may be employed.

In Crelle's Journal, Vol. LXI. there is an article by Professor Boole, entitled Ueber die partielle Differentialgleichung zweiter Ordnung Rr + Ss + Tt + U (s2 — rt) = V.

The article is dated 1862; it occupies pages 309-333 of the volume.

Among Professor Boole's manuscripts I found a memoir very closely resembling the article in Crelle's Journal; it

would appear that the memoir was drawn up with a view to publication in the Transactions of some English Scientific Society, and that this design was afterwards abandoned in favour of the article in Crelle's Journal.

After some hesitation I have resolved to print this memoir. Even if the memoir had been identical with the article in Crelle's Journal it would have been convenient to the English reader to be able to avail himself of the investigations; and the memoir contains remarks which do not occur in the article, and which are interesting in connexion with the history of the subject. There is some repetition of matter which has already been given in Chapter XXVIII.; but I was unwilling to impair the completeness of the memoir by abridgment or omission. Accordingly the memoir forms the next Chapter of the present volume.

In Article 2 of the next Chapter will be found the process to which there is an allusion towards the end of Article 4 of Chapter XXVIII.

It is obvious that the subject of partial differential equations of the second order was much studied by Professor Boole. The chronological order of his writings on the subject appears to be as follows:

1. Chapter XV. of the first edition of his work.

2. The article in the Bulletin of St Petersburg.

3. The memoir which forms Chapter XXIX. of the present volume.

4. The article in Crelle's Journal.

5. The Chapter XXVIII. of the present volume.]