Theory of Ideas, before you have gone through a proper intellectual discipline. The impulse which urges you to such speculations is admirable-is divine. But you must exercise yourself in reasoning which many think trifling, while you are yet young; if you do not, the truth will elude your grasp." Socrates asks submissively what is the course of such discipline: Parmenides replies, "The course pointed out by Zeno, as you have heard." And then, gives him some instructions in what manner he is to test any proposed Theory. Socrates is frightened at the laboriousness and obscurity of the process. He says, "You tell me, Parmenides, of an overwhelming course of study; and I do not well comprehend it. Give me an example of such an examination of a Theory." "It is too great a labour," says he, "for one so old as I am." "Well then, you, Zeno," says Socrates," will you not give us such an example?" Zeno answers, smiling, that they had better get it from Parmenides himself; and joins in the petition of Socrates to him, that he will instruct them. All the company unite in the request. Parmenides compares himself to an aged racehorse, brought to the course after long disuse, and trembling at the risk; but finally consents. And as an example of a Theory to be examined, takes his own Doctrine, that All Things are One, carrying on the Dialogue thenceforth, not with Socrates, but with Aristoteles (not the Stagirite, but afterwards one of the Thirty), whom he chooses as a younger and more manageable respondent.

So.

The discussion of this Doctrine is of a very subtle kind, and it would be difficult to make it intelligible to a modern reader. Nor is it necessary for my purpose to attempt to do It is plain that the discussion is intended seriously, as an example of true philosophy; and each step of the process is represented as irresistible. The Respondent has nothing to say but Yes; or No; How so? Certainly; It does appear; It does not appear. The discussion is carried to a much greater length than all the rest of the Dialogue; and the result of the reasoning is summed up by Parmenides thus: "If One exist, it is Nothing. Whether One exist or do not exist, both It and Other Things both with regard to Themselves and to Each other, All and Everyway are and are not, appear and appear not." And this also is fully assented to; and so the Dialogue ends.

I shall not pretend to explain the Doctrines there examined that One exists, or One does not exist, nor to trace their consequences. But these were Formulæ, as familiar in the Eleatic school, as Ideas in the Platonic; and were undoubtedly regarded by the Megaric contemporaries of Plato as quite worthy of being discussed, after the Theory of Ideas had been overthrown. This, accordingly, appears to be the purport of the Dialogue; and it is pursued, as we see, without any bitterness towards Socrates or his disciples; but with a persuasion that they were poor philosophers, conceited talkers, and weak disputants.

The external circumstances of the Dialogue tend, I conceive, to confirm this opinion, that it is not Plato's. The Dialogue begins, as the Republic begins, with the mention of a Cephalus, and two brothers, Glaucon and Adimantus. But this Cephalus is not the old man of the Piræus, of whom we have so charming a picture in the opening of the Republic. He is from Clazomenæ, and tells us that his fellow-citizens are great lovers of philosophy; a trait of their character which does not appear elsewhere. Even the brothers Glaucon and Adimantus are not the two brothers of Plato who conduct the Dialogue in the later books of the Republic: so at least Ast argues, who holds the genuineness of the Dialogue.

This

Glaucon and Adimantus are most wantonly introduced; for the sole office they have, is to say that they have a half-brother Antiphon, by a second marriage of their mother. No such half-brother of Plato, and no such marriage of his mother, are noticed in other remains of antiquity. Antiphon is represented as having been the friend of Pythodorus, who was the host of Parmenides and Zeno, as we have seen. And Antiphon, having often heard from Pythodorus the account of the conversation of his guests with Socrates, retained it in his memory, or in his tablets, so as to be able to give the full report of it which we have in the Dialogue Parmenides*. To me, all this looks like a clumsy imitation of the Introductions

to the Platonic Dialogues.

I say nothing of the chronological difficulties which arise from bringing Parmenides and Socrates together, though they are considerable; for they have been explained more or less satisfactorily; and certainly in the Theatetus, Socrates is represented as saying that he when very young had seen Parmenides who was very old t. Athenæus, however ‡, reckons this among Plato's fictions. Schleiermacher gives up the identification and relation of the persons mentioned in the Introduction as an unmanageable story.

I may add that I believe Cicero, who refers to so many of Plato's Dialogues, nowhere refers to the Parmenides. Athenæus does refer to it; and in doing so blames Plato for his coarse imputations on Zeno and Parmenides. According to our view, these are hostile attempts to ascribe rudeness to Socrates or to Plato. Stallbaum acknowledges that Aristotle nowhere refers to this Dialogue.

In the First Alcibiades, Pythodorus is mentioned as having paid 100 minæ to Zeno for his instructions (119 A). † p. 183 e. Deipn. XI. c. 15, p. 105.

VI. On the Discontinuity of Arbitrary Constants which appear in Divergent Developments. By G. G. STOKES, M.A., D.C.L., Sec. R.S., Fellow of Pembroke College, and Lucasian Professor of Mathematics in the University of Cambridge.

[Read May 11, 1857.]

IN a paper "On the Numerical Calculation of a class of Definite Integrals and Infinite Series," printed in the ninth volume of the Transactions of this Society, I succeeded in developing the integral (cos(w3 - mw) dw in a form which admits of extremely easy

2

numerical calculation when m is large, whether positive or negative, or even moderately large. The method there followed is of very general application to a class of functions which frequently occur in physical problems. Some other examples of its use are given in the same paper; and I was enabled by the application of it to solve the problem of the motion of the fluid surrounding a pendulum of the form of a long cylinder, when the internal friction of the fluid is taken into account *.

They satisfy

These functions admit of expansion, according to ascending powers of the variables, in series which are always convergent, and which may be regarded as defining the functions for all values of the variable real or imaginary, though the actual numerical calculation would involve a labour increasing indefinitely with the magnitude of the variable. certain linear differential equations, which indeed frequently are what present themselves in the first instance, the series, multiplied by arbitrary constants, being merely their integrals. In my former paper, to which the present may be regarded as a supplement, I have employed these equations to obtain integrals in the form of descending series multiplied by exponentials. These integrals, when once the arbitrary constants are determined, are exceedingly convenient for numerical calculation when the variable is large, notwithstanding that the series involved in them, though at first rapidly convergent, became ultimately rapidly divergent.

The determination of the arbitrary constants may be effected in two ways, numerically or analytically. In the former, it will be sufficient to calculate the function for one or more values of the variable from the ascending and descending series separately, and equate the results. This method has the advantage of being generally applicable, but is wholly devoid of elegance. It is better, when possible, to determine analytically the relations between the

Camb. Phil. Trans. Vol. IX. Part II.

VOL. X. PART I.

14

arbitrary constants in the ascending and descending series. In the examples to which I have applied the method, with one exception, this was effected, so far as was necessary for the physical problem, by means of a definite integral, which either was what presented itself in the first instance, or was employed as one form of the integral of the differential equation, and in either case formed a link of connexion between the ascending and the descending series. The exception occurs in the case of Mr Airy's integral for m negative. I succeeded in determining the arbitrary constants in the divergent series for m positive; but though I was able to obtain the correct result for m negative, I had to profess myself (p. 177) unable to give a satisfactory demonstration of it.

But though the arbitrary constants which occur as coefficients of the divergent series may be completely determined for real values of the variable, or even for imaginary values with their amplitudes lying between restricted limits, something yet remains to be done in order to render the expression by means of divergent series analytically perfect. I have already remarked in the former paper (p. 176) that inasmuch as the descending series contain radicals which do not appear in the ascending series, we may see, a priori, that the arbitrary constants must be discontinuous. But it is not enough to know that they must be discontinuous; we must also know where the discontinuity takes place, and to what the constants change. Then, and not till then, will the expressions by descending series be complete, inasmuch as we shall be able to use them for all values of the amplitude of the variable.

I have lately resumed this subject, and I have now succeeded in ascertaining the character by which the liability to discontinuity in these arbitrary constants may be ascertained. I may mention at once that it consists in this; that an associated divergent series comes to have all its terms regularly positive. The expression becomes thereby to a certain extent illusory; and thus it is that analysis gets over the apparent paradox of furnishing a discontinuous expression for a continuous function. It will be found that the expressions by divergent series will thus acquire all the requisite generality, and that though applied without any restriction as to the amplitude of the variable they will contain only as many unknown constants as correspond to the degree of the differential equation. The determination, among other things, of the constants in the development of Mr Airy's integral will thus be rendered complete.

1. Before proceeding to more difficult examples, it will be well to consider a comparatively simple function, which has been already much discussed. As my object in treating this function is to facilitate the comprehension of methods applicable to functions of much greater complexity, I shall not take the shortest course, but that which seems best adapted to serve as an introduction to what is to follow.

The integral and the series are both convergent for all values of a, and either of them completely defines u for all values real or imaginary of a. We easily find from either the integral or the series

This integral or series like the former gives a determinate and unique value to u for any assigned value of a real or imaginary. Both series, however, though ultimately convergent, begin by diverging rapidly when the modulus of a is large. For the sake of brevity I shall hereafter speak of an imaginary quantity simply as large or small when it is meant that its modulus is large or small.

2. In order to obtain u in a form convenient for calculation when a is large, let us seek to express u by means of a descending series. We see from (2) that when the real part of a2

is positive, the most important terms of the equation are 2au and 2, and the leading term of the development is a'. Assuming a series with arbitrary indices and coefficients, and determining them so as to satisfy the equation, we readily find

This series can be only a particular integral of (2), since it wants an arbitrary constant. To complete the integral we must add the complete integral of

This expression might have been got at once from (3) by integration by parts. It remains to determine the arbitrary constant. C.

3. The expression (1) or (3) shews that u is an odd function of a, changing sign with a. But according to (4) u is expressed as the sum of two functions, the first even, the second odd, unless C = 0, in which case the even function disappears. But since, as we shall presently see, the value of C is not zero, it must change sign with a.

a = p (cos + √ - 1 sin 0).

Let

Since in the application of the series (4) it is supposed that p is large, we must suppose a to change sign by a variation of 0, which must be increased or diminished (suppose increased) by π. Hence, if we knew what C was for a range of 0, suppose from a to = a + π, we should know at once what it was from = a + to 0 = a + 2π, which would be sufficient