is completely determined. (See Dirichlet's memoir on the Arithmetical Progression, sect 7, in the Berlin Memoirs for 1837.)

78. Primitive Roots of the Powers of Complex Primes.-Dirichlet has shown* that, in the theory of complex numbers of the form a+bi, the powers of primes of the second species (see art. 25) have primitive roots; in fact, if a+bi be such a prime, and N (a+bi)=a2+b2=p, every primitive root of pm is a primitive root of (a+bi)m. On the other hand, if q be a real prime of the form 4n+3, qm has no primitive roots in the complex theory. For in general, if M be any complex modulus, and Max B cr a, b, c, .. being different complex primes, and if A=N (a), B=N (b), C=N (c), etc., the number of terms in a system of residues prime to M, is A-1 (A-1) B-1 (B—-1) C11 (C−1)........... And if we denote this number by (M), every residue prime to M will satisfy the congruence

x (M) = 1, mod M,

2m-2

...

1,

which here corresponds to Euler's extension of Fermat's theorem. If M=qm, this congruence becomes a ·2 (92-1) 1, mod qm; but it is easily shown that every residue prime to qm satisfies the congruence x gm-1 (q2—1) mod qm; i. e., qm has no primitive roots, because the exponent qm-1 (q2 — 1 ) is a divisor of, and less than, q2(m-1) (q2-1). Nevertheless two numbers y and y', can always be assigned, of which one appertains to the exponent qm-1 (q-1) and the other to the exponent qm-1, and which are such that no power of either of them can become congruous to a power of the other, mod qm, without becoming congruous to unity; from which it will appear that every residue prime to qm may be represented by the formula y y', if we give to a all values from 0 to (q2—1) qm-1 −1 inclusive, and to y all values from 0 to qm-1-1 inclusive.

The corresponding investigations for other complex numbers besides those of the form a+bi have not been given.

We here conclude our account of the Theory of Congruences. The further continuation of this Report will be occupied with the Theories of Quadratic and other Homogeneous Forms.

་

Additions to Part I.

Art. 16. Legendre's investigation of the law of reciprocity (as presented in the Théorie des Nombres,' vol. i. p. 230, or in the Essai,' ed. 2, p. 198) is invalid only because it assumes, without a satisfactory proof, that if a be a given prime of the form 4n+1, a prime b of the form 4n+3 can always

be assigned, satisfying the equation (#)=-1. M. Kummer (in the Memoirs

of the Academy of Berlin for 1859, pp. 19, 20) says that this postulate is easily deducible from the theorem demonstrated by Dirichlet, that every arithmetical progression, the terms of which have no common divisor, contains prime numbers. It would follow from this, that the demonstration of Legendre (which depends on a very elegant criterion for the resolubility or irresolubility of equations of the form ax2+by+cz2=0) must be regarded as rigorously exact (see, however, the "Additamenta" to arts. 151, 296, 297 of the Disq. Arith.). In the introduction to the memoir to which we have just referred, the reader will find some valuable observations by M. Kummer on the principal investigations relating to laws of reciprocity.

*See sect. 2 of the memoir, Untersuchungen über die Theorie der complexen Zahlen, in the Berlin Memoirs for 1841.

Art. 20. Dirichlet's demonstration of the formulæ (A) and (A') first appeared in Crelle's Journal, vol. xvii. p. 57. Some observations in this paper on a supposed proof of the same formula by M. Libri (Crelle, vol. ix. p. 187) were inserted by M. Liouville in his Journal, vol. iii. p. 3, and give rise to a controversy (in the Comptes Rendus, vol. x.) between MM. Liouville and Libri. The concluding paragraphs of Dirichlet's paper contain the application of the formulæ (A) and (A') to the law of reciprocity (Gauss's fourth demonstration).

Art. 22. From a general theorem of M. Kummer's (see arts. 43, 44 of this

A-1

Report), it appears that the congruence r2=(−1) 2 λ, mod q, is or is not

A-1

resoluble, according as q 2 =+1, or =— —1, mod λ,—a result which implies the theorem of quadratic reciprocity. This very simple demonstration (which is, however, only a transformation of Gauss's sixth) appears first to have occurred to M. Liouville (see a note by M. Lebesgue in the Comptes Rendus, vol. li. pp. 12, 13).

Art. 24. A note of Dirichlet's, in Crelle, vol. lvii. p. 187, contains an elementary demonstration of Gauss's criterion for the biquadratic character of 2. From the equation p=a+b2, we have (a+b)2=2ab, mod p, and hence (a+b)+(p−1)=24 (p−1) at (p −1) bi (p−1) = (2ƒ) $(P−1) at (p-1), or, which is the same thing,

(a+b)

(A)

because p=b2, mod a;

serving that 2p=(a+b)2+(a−b)2,

8 =ƒt (p-1)+ab,

since f+1=0, mod p. Substituting these values in the equation (A), we find 24(p−1) =ƒtab, mod p, which is in fact Gauss's criterion.

Art. 25. In the second definition of a primary number, for "b is uneven," read "b is even." Although this definition has been adopted by Dirichlet in his memoir in Crelle's Journal, vol. xxiv. (see p. 301), yet, in the memoir "Untersuchungen über die complexen Zahlen" (see the Berlin Memoirs for 1841), sect. 1, he has preferred to follow Gauss.

Art. 36. In the algorithm given in the text, the remainders p2, P ̧... are all

uneven; and the computation of the value of the symbol (2) is thus rendered

independent of the formula (iii) of art. 28. The algorithm given by Eisenstein is, however, preferable, although the rule to which it leads cannot be expressed with the same conciseness, because the continued fraction equivalent to Po terminates more rapidly when the remainders are the least possible, and not necessarily uneven.

Pi

Art. 37. In the definition of a primary number, for "a=+1," read "a=-1." But, for the purposes of the theory of cubic residues, it is simpler to consider the two numbers (a+bp) as both alike primary (see arts. 52 and 57).

Art. 38. Jacobi's two theorems cannot properly be said to involve the

cubic law of reciprocity. If (2)=1, it will follow from those theorems that

2

or p2. It is remarkable that these theorems, "formâ genuinâ quâ inventa sunt," may be obtained by applying the criteria for the resolubility or irresolubility of cubic congruences (art. 67) to the congruence r3-3 Ar—λM=0, mod q (art. 43), which, by virtue of M. Kummer's theorem (art. 44), is resoluble or irresoluble according as q is or is not a cubic residue of X.

On the Performance of Steam-Vessels, the Functions of the Screw, and the Relations of its Diameter and Pitch to the Form of the Vessel. By Vice-Admiral MOORSOм.

(A communication ordered to be printed among the Reports.)

In this the fourth paper which I now lay before the British Association, it may be desirable to recapitulate the points I have brought into issue, and for the determination of which, data, only to be obtained by experiments, are still wanting, viz.

1. There is no agreed method by which the resistance of a ship may be calculated under given conditions of wind and sea.

2. The known methods are empirical, approximate only, and imply smooth water and no wind.

3. The relations in which power and speed stand to form and to size are comparatively unknown.

4. The relations in which the direct and resultant thrust stand to each other in any given screw, and how affected by the resistance of the ship, are undetermined.

In order to resolve these questions, specific experiments are needed, and none have yet been attempted in such manner as to lead to any satisfactory result.

The Steam Ship Performance Committee of the British Association have pressed upon successive First Lords of the Admiralty, the great value to the public service which must ensue if the following measures were taken, viz.—

1. To determine, by specific experiment, the resistance, under given conditions, of certain vessels, as types; and, at the same time, to measure the thrust of the screw.

2. To record the trials of the Queen's ships, so that the performance in smooth water may be compared with the performance at sea, both being recorded in a tabular form, comprising particulars, to indicate the characteristics of the vessel, of the engine, of the screw, and of the boiler.

Hitherto nothing has come of these representations.

In the paper read last year at Aberdeen, I showed, in the case of Lord Dufferin's yacht 'Erminia,' how the absence of admitted laws of resistance interfered with the adjustment of her screw, and how, therefore, as a matter of precaution, a screw was provided capable of a thrust beyond what the vessel required.

I also showed, in the case of the Duke of Sutherland's yacht Undine,' how her screw, from being too near the surface of the water, lost a large portion of the thrust due to its size and proportions. In other words, a screw capable of giving out a resultant thrust in sea water of 5022 lbs., at a speed of

vessel of 9.26 knots an hour, did actually give out only 3805 lbs. That is to say, the effect produced was the same as if that screw had worked in a fluid whose weight was about 48 lbs. per cubic foot instead of 64 lbs.

I am now about to exhibit some other examples from among Her Majesty's ships of war.

The questions now before us are―

1. The resistance of the hull below the water-line in passing through the water, and of the upper works, masts, rigging, &c., passing through the air, the weather being calm, and the water smooth.

2. The relation in which the thrust of the screw stands to this resistance. [The Admiral here gave certain results from the Marlborough,' the 'Renown,' and the 'Diadem,' and proposed that a specific issue should be tried by means of the 'Diadem.']

What would I not give, he observed, for some well-conducted experiments to determine this beautiful problem of the laws which govern the action of the screw in sea-water! It is a problem not only interesting to science, but fraught with valuable results in the economical and efficient application of the screw propeller.

After commenting on the performances of the U. S. corvette 'Niagara,' the Admiral observed, I have no means of forming a very definite opinion as to how she will stay under low sail in a sea-way, how she will wear, how scud in a following sea, or how stand up under her sails, or whether her statical stability be too much or too little, or how the fore and after bodies are balanced. These are points to be determined, not by the mere opinion of seamen--for a sailor will vaunt the qualities of his ship even as a lover the charms of his mistress-but by careful records of performances in smooth water and at sea, and a comparison of such performances with calculated results from drawings beforehand. Let a return of such things be annually laid before the House of Commons-we shall then know whether we are getting money's worth for our money; and also we should receive all the benefits of public criticism towards improvement. We should not then allow defects to be stereotyped, till chronic blemishes are turned into beauties, or, if not so, then defended as things that cannot be remedied.

I have now completed the task which four years ago I imposed on myself. Beginning with simple elementary principles, and ending with minute practical details, I have, as I conceive, shown the process by which the improvement of steam-ships must be carried on.

More than one hundred years ago scientific men, able mathematicians, showed the physical laws on which naval architecture must rest. A succession of able men have shown how those laws affect various forms of floating bodies. Experiments have been made with models to determine the value of the resistance practically. With the exception of some experiments of Mr. Scott Russell, I am not aware that any have been made with vessels approaching the size of ships to determine the relations of resistance to power, whether wind or steam.

Ships have been improved, and modifications of form have been arrived at by a long painstaking tentative process. The rules so reached for sailing ships have been superseded by steam, and we are still following the same tedious process, in order to establish new rules for the application of steam power.

I think the history of naval architecture shows that it is not an abstract science, and that its progress must depend on the close observation and correct record of facts; on the careful collating, and scientific comparing of such facts, with a view to the induction of general laws. Now, is there anywhere such observing, recording, collating, and comparing? and still more, is there such inducting process?

I can find no such thing anywhere in such shape that the public can judge it by its fruits.

We are now in full career of a competition of expenditure, and England has no reason to flinch from such an encounter, unless her people should tire of paying a premium of insurance upon a contingent event that never may happen; and if it should happen without our being insured, might not cost as much as the aggregate premiums. Tire they will, sooner or later, but they are more likely to continue to pay in faith and hope, if they had some confidence that their money is not being spent unnecessarily.

There is now building at Blackwall the 'Warrior,' a ship to be cased with 44-inch plates of iron, whose length at water-line is 380 feet, breadth 58 feet, intended draught of water (mean) 25 feet, area of section 1190 square feet, and displacement about 8992 tons, and she is to have engines of 1250 nominal horse-power.

Is there any experience respecting the qualities and performance of such a ship? Anything to guide us in reasoning from the known to the unknown? Do the performances of the Diadem,' Mersey,' and 'Orlando,' inspire confidence? Where are the preliminary experiments?

Before any contract was entered into for the construction of the Britannia Bridge, a course of experiments was ordered by the Directors, which cost not far short of £7000, and it was well expended. It saved money, and perhaps prevented failure. This ship must cost not less than £400,000, and may cost a good deal more when ready for sea. But there is another of similar, and two others building, of smaller size. What security is there for their success?

The conditions which such a ship as the 'Warrior' must fulfil in order to justify her cost are deserving of some examination. The formidable nature of her armament, as well as her supposed impregnability to shot, will naturally lead other vessels to avoid an encounter. She must therefore be of greater speed than other ships of war. To secure this, it is essential that her draught of water should be the smallest that is compatible both with stability and steadiness of motion, and that she should not be deeper than the designer intended. To ensure steadiness it is necessary, among other things, that in rolling, the solids, emerged and immersed, should find their axis in the longitudinal axis of the ship. To admit of accurate aim with the guns, her movement in rolling should be slow and not deep. Every seaman knows how few ships unite these requisites.

It is not quite safe to speculate on the 'Warrior's' speed; nevertheless I will venture on an estimate, such as I have stated in the case of the Great Eastern,' whose smooth-water speed I will now assume to be 154 knots, as before estimated, with 7732 horse-power, when her draught of water is 23 feet, her area of section, say 1650 square feet, and her displacement about 18,588 tons. The speed of the 'Warrior' in smooth water ought not to be less than 16 knots, in order that she may force to action unwilling enemies whose speed may be 13 to 14 knots.

The question I propose is the power to secure a smooth-water speed of 16 knots.

Reducing the Great Eastern' to the size of the Warrior,' and applying the corrections for the difference of speed of knot, and for their respective coefficients of specific resistance ·0564 and '07277, the horse-power for 16 knots is 7543.

Raising the Niagara' to the size of the 'Warrior,' and applying the corrections for the difference of speed between 10′9 and 16 knots, and for their respective coefficients of specific resistance 0797 and 07277, the horse-power to give the Warrior' a smooth-water speed of 16 knots is 7867, being an excess over the estimate from the Great Eastern' of 324 horse-power.