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 U nor V vanishes.

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

nate from the general system (22) and

dF

dF

in succes

dq dp

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

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 Petersbourg, 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

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

dz

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.]

CHAPTER XXIX.

ON THE SOLUTION OF THE PARTIAL DIFFERENTIAL EQUATION Rr + Ss + Tt + U (s2 — rt) = V, IN WHICH R, S, T, U, V ARE GIVEN FUNCTIONS OF x, y, z, P, Q•

1. THE equation, the theory of the solution of which I propose to consider in this paper, is remarkable from its connexion with Geometry. If the equation of a surface contain three constants which vary as parameters in subjection to any two conditions connecting them, the generated envelope will satisfy a partial differential equation of the above form. In other words any envelope of the surface

F(x, y, z, a, b, c) = 0

formed by the variation of a, b, c in subjection to two connecting conditions

P1 (a, b, c) = 0, $2 (a, b, c) = 0

is necessarily an integral of a partial differential equation of the form given above.

Now this theorem is the more important, because it is only when three parameters in the equation of a surface vary in subjection to two relations that the envelope possesses, irrespectively of the form of the connecting relations, any definite character. If there be but one connecting relation it is possible to determine that relation so as to make the envelope assume the form of any surface whatever, and therefore the possible system of envelopes is in such case

B. D. E. II.

10

unlimited. If there be three connecting relations the parameters become absolutely constant and no envelope exists.

The partial differential equation

Rr + Ss + Tt + U (s2 — rt) = V

is remarkable also as including all the cases in which a partial differential equation of the second order admits a first integral of the form

u = f(v),

u and v being definite functions of x, y, z, p, q, and ƒ (v) arbitrary in form.

Neither of these statements is sufficiently general to constitute a theory of the genesis of the partial differential equation under consideration, but the second one is more general than the first, and is indeed sufficiently so to serve as the ground of an investigation which connects the solution of the equation in all cases with the satisfaction of a system of simultaneous ordinary differential equations of the first order and degree. And this is the ground upon which the method of the paper will rest. I propose to shew, 1st that the solution of the given equation on the assumption that a first integral of the form u=f (v) exists requires the satisfaction of a system of two partial differential equations of the first order and second degree; 2ndly that this system may be resolved into four systems, each consisting of two partial differential equations of the first order and first degree, two of which systems are irrelevant and the other two relevant; 3rdly that the solution of the two relevant systems ultimately depends on the solution of a system of ordinary differential equations of the first order, and that from these ordinary differential equations the given equation of the second order may be deduced independently of the assumption above mentioned. I shall also discuss the theory of the second integration. And I shall exemplify another method of solution connected by a remarkable law of reciprocity with the above method.