meter resistance is greater than the battery resistance (a condition which should always be fulfilled in testing telegraph-lines), the galvanometer should be made to connect the junction of the two greater resistances with that of the two lesser, thus:

Now, considering the question of what resistance practically to give a galvanometer intended to be used in a Wheatstone's bridge for testing telegraph-lines, the condition of maximum sensibility becomes

Supposing the branches a and b to each consist of three coils of 10, 100, and 1000 ohms resistance respectively, and w to be variable between 1 ohm and 10,000 ohms, and putting*

1866.

Electrician's Office, Alipore,

November 21, 1873.

See Mr. Schwendler's paper in the Philosophical Magazine for May

V. Further Discussion of the Analytical Principles of Hydrodynamics, in Reply to Mr. Moon. By Professor CHALLIS, M.A., LL.D., F.R.S., F.R.A.S.*

IN

N vol. xxxvi. of the Philosophical Magazine (p. 117), and in the Number for September 1873 (p. 247), Mr. Moon has adduced an argument which, as expressed in the later publication, is as follows:-" For motion in one direction it is axiomatic that we must have in every fluid, in every case of motion,

p=f(x, t), p=f2(x, t), v=f3(x, t);

whence it follows that, for every value of x and t for which each of the foregoing equations represents a substantive relation between the variables, we shall have p= funct. (p, v).”

I have admitted the truth and the comprehensiveness of this conclusion, considering it to be a legitimate inference from the axiom that all physical relations expressible by functions of space and time are comprehended by the processes of abstract calculation (see Phil. Mag. for August 1873, pp. 160 & 165).

Also in my reply in the October Number (p. 310), I have extended the argument to space of three dimensions, and obtained the general equation F(p, p, u, v, w)=0.

I have, besides, ascertained that the values of p, p, v obtained by Mr. Moon in vol. xxxvi. (p. 124), and reproduced in the Number for December 1873 (p. 448), satisfy the differential equation

Having made these statements, I am prepared to admit that Mr. Moon has rightly urged (in the December Number, p. 447), in opposition to an assertion I had made, that the assigning of a relation between p and v in the equation p=f(p, v) does not simply define the fluid, but imposes also conditions on the motion existing in it, and that an analogous remark applies to the equation F(p, p, u, v, w)=0. Let us, however, inquire what inferences relative to this point are deducible from Mr. Moon's equations.

By a solution of the equation (1) he obtains for determining p the equation

a being an arbitrary constant, and the form of the function x being arbitrary. Putting on the left-hand side of this equation

s(p, v) for p, it will be seen, since x(+) is arbitrary in form

* Communicated by the Author.

and value, that v is an arbitrary function of p, and, by consequence, on substituting for v in f(p, v), that p is also an arbitrary function of p. Hence, by Mr. Moon's own reasoning, there may be an unlimited number of different relations between p and p, each defining a different fluid; and evidently among these the ordinary relation p=a'p must be included.

dy

Again, since v = and =D (dy)

dt

p=

shown that v is an arbitrary function of p, it follows that

By differentiating equation (2) with respect to t, we obtain

(2)

(3)

which equation has the same generality as that from which it was derived.

The equations (2) and (3) are given by Mr. Earnshaw at the commencement of a paper "On the Mathematical Theory of Sound," contained in the Philosophical Transactions (vol. cl. p. 133), which is chiefly devoted to the discussion of the particular case in which,

dy

dx

D

being always equal to we have

ρ

(4)

this equation being that which (3) becomes on the hypothesis of Boyle's law. Now the equation (1) is identical with the equation (4) on the same hypothesis. For since p=D(1), and p=ap, it follows that

-1 dy

dx

day
dx2

= a2

-D.

dx

dx

dy2

dx2

;

dp

whence by substituting for

dx

in (1) the equation (4) results.

At the end of his paper Mr. Earnshaw briefly indicates the mode of treatment of the case in which

this form of the equation (3) is derivable from Mr. Moon's equation p=fp, v) and the equation (1). From the value of p we get by differentiation

and is therefore equivalent to the above-cited equation,

dy

Since dy = F(d).
F(1), the last equation may be assumed to be iden-

dt

tical with the equation (3), which, as Mr. Earnshaw has justly remarked, "can be made to coincide with any dynamical equaday dzy tion in which the ratio of to can be expressed in terms di2 dx2

It has thus, I think, been sufficiently proved that Mr. Moon's hydrodynamical researches are founded on differential equations which are really the same as those employed by Mr. Earnshaw, and differ only in the process of investigation and form of expression, and should consequently lead to the same results. This identity indicates that the two mathematicians have argued correctly, although by different processes, from the same princi

ples, which, as certainly in the one case as in the other, are the commonly received principles of hydrodynamics. It is therefore logically impossible that any argument against those principles can be drawn from Mr. Moon's reasoning or his results, inasmuch as the reasoning has no other basis to stand upon.

I think it right to add that I have advanced the foregoing argument only as an argumentum ad hominem, intended to show that, even admitting Mr. Moon's premises and his reasoning, no conclusion can be drawn therefrom contradictory to the principles of hydrodynamics. My reason for making this statement here is that I have repeatedly, in this Journal and elsewhere, asserted that I cannot accept either the equation (1) or the equation (4) as a true hydrodynamical equation-not because the principles on which these equations rest are untrue, but because they require to be supplemented. At the same time I can admit the truth and the importance of the equations p=f(p, v) and F(p, p, u, v, w) =0, considering them to be necessary for completing the analytical principles of hydrodynamics. Experiment can establish the truth of the equation p=a2p only for fluid at rest, whereas by means of these equations it can be shown to be legitimate to assume the same equation to hold good for fluid in motion.

Having given these explanations I think it needless to pursue the subject at greater length, and accordingly shall decline entering upon any further discussion.

Cambridge, December 16, 1873.

VI. On the Electromotive and Thermoelectric Forces of some Metallic Alloys in contact with Copper. By A. F. SUNDELL, Docent at the University of Helsingfors*.

THE

THE remarkable thermoelectric properties of metallic alloys discovered by the investigations of Seebeck, Rollmann, and other physicists permitted it to be supposed that interesting phenomena must also occur in relation to the power of these bodies, in contact with each other or with unmixed metals, to act in exciting electricity. At the desire of Professor Edlund, therefore, who has determined by a new method the electromotive forces of a greater number of metals in contact with copper, and compared them with the thermoelectric forces †, I under

Translated from a separate impression, communicated by the Author, from Poggendorff's Annalen, vol. cxlix. pp. 144-170.

+ Kongl. Svenska Vet.-Akademiens Handlingar, vol. ix. (1871) No. 14; Phil. Mag. Jan. 1871, p. 18; Pogg. Ann. vol. cxliii. pp. 404 & 534. A preliminary investigation will be found in Efversigt af K.Vet.-Akad. Förhandlingar 1870, p. 3; Pogg. Ann. vol. cxl. p. 435.