morrow. To what extent this vitiating action operates may be inferred from the fact that I have never been able to obtain from the gas of our pipes an illuminating power equal to the minimum of that reported by the engineer of the gas company. In my trials the power has varied from that of 13 candles down as low as that of 9 candles, instead of ranging from 14 to 17 candles.

This difference is perfectly intelligible if we assume the last quantities to represent the value of the gas when first made, and my results to represent its value as delivered to the consumer. In conclusion I would merely add that the difficuly suggests its own remedy. And that would be to have a standard of quality established by the proper authorities, taking the illuminating power as the basis of the calculation, and then to have the requirements of such standard insured by a nightly examination, if necessary on the part of some one entirely disconnected with the manufacture. In other words the photometer can be made as available and as valuable to the consumer of gas as the hydrometer is to the spirit merchant; as he distinguishes with his instrument in any mixture, between the spirit he wishes to buy and the water he is unwilling to pay for, so the consumer of gas can distinguish with the photometer between the true illuminating material and the worthless heat producing gases, hydrogen and light curburetted hydrogen, that make up the bulk of the ordinary coal gas.

ART. XIV.—On the Dynamical Condition of the Head of a Comet; by Professor W. A. NORTON.

It is proposed, in the present article, to give the mathematical theory of the development of the nebulous envelope of the head, and the tail of a comet from the nucleus; under the combined action of a repulsive force exerted by the nucleus, and a repulsive force exerted by the sun-each of these forces being supposed to vary inversely as the square of the distance from the centre of the repelling mass. So far as I am aware, no attempt has hitherto been made to develope the idea of a dynamical condition of the head of a comet into a mathematical theory, based upon precise numerical laws.

The hypothesis that a projectile force is in operation, combined with a repulsive action, or even with a gravitating force only, will also be briefly considered.

Let us first suppose cometic matter to be expelled from all points of the surface of the nucleus, on the side toward the sun, and in directions normal to the surface, regarded as spherical. As the nucleus is very small, in comparison with the nebulosity

of the head, the error will be slight if we regard it as a mere point, and conceive the nebulous matter to be repelled in all directions from this point. At the same time it must be observed that, for each expelled particle, the central point of repulsion is below the point of emission a distance equal to the radius of the nucleus. Again, as the rectilinear dimensions of the head of a comet are small as compared with its distance from the sun, we may, without material error for our present purpose, regard the repulsive force of the sun as constant.

Let N (fig. 1) be the nucleus, regarded as a point, NS the direction of the sun, and AB a line perpendicular to NS, which

we will regard as the line of demarcation between the head and tail. Suppose a particle to be emitted in the direction NZ, and let the angle of projection, ZNB = a. Also let p= acceleration due to excess of repulsion of nucleus over its attraction, at the surface of the nucleus; k= opposing acceleration, from sun's repulsive force; and r radius of nucleus, or the distance of the point of emission from the centre of repulsion, wherever this may be. We will first undertake an approximate investigation, by disregarding the effect of the recess of the particle from the line NZ upon the repulsive force of the nucleus. This amounts to supposing that the centre of repulsion moves along a line perpendicular to NZ, at the same rate that the particle recedes from NZ.

Decompose the sun's repulsive force into two components,the one along NZ, and the other perpendicular to it. The former will be ksina, and the latter cosa. Denoting by the distance passed over by the receding particle, in the direction NZ, in any interval of time; and by v the velocity at that distance, we have,

If we suppose the initial velocity to be zero, v=0, when z=r,

and

v2

Whence, pra (-)- k sin a (z − r) = ""; or, pr2("-") — ksina

2

(1)

Let Z= greatest distance passed over in the direction NZ, and we have

pr
Z

k sin a=0; or, Z=

pr
k sin a

(2)

This value of z is the distance NZ, X being the point where the orbit is tangent to the line ZX, perpendicular to NZ. Putting a=90°, we get for the distance to which a particle will recede from the nucleus, when emitted in the direction NS,

pr But, by equ. (2), Zsina = =- = H; also Z sin a = Z sin N ZV k =NV; hence NV=H, and the point Z will fall on VT, drawn through V, at the distance H from N, and perpendicular to NS. To find X, the point of tangency, resume equ. (1); and, since remote from the nucleus r may be neglected in comparison with

dz2

z, we have

dt2

1

dz

√2pr-ksina.z

This formula is quite accurate for determining any portion of the interval of time sought, for the beginning of which z is large in comparison with r, and as the motion is far more rapid in the vicinity of the nucleus than at a distance from it, we may obtain pretty nearly the whole interval of time, from N to X, by supposing z, at the beginning, to be several times r; but, on this

supposition the formula gives nearly the same value, as when z is made equal to zero. Therefore let t=0, when z=0, and we

√2pr

k sin a

To verify this value of T, I have made another calculation, by dividing the time into two intervals. The first extends to the instant of time when the distance becomes equal to the

part

of the whole distance, Z; during which the motion may be regarded as very nearly uniform, with an average velocity equal to pr. The calculation for the remaining interval was made by the above equation (a). The result obtained is T=

√2pr (1+). This differs from the above determination, by

k sin a

only

85'

5

in the case of the comet of 1811, and about

instance of Donati's Comet.

To find ZX=X, we have X=kcosa. T2=kcos a

k2 sin 2a

cot ZXN; hence_ZXN=a=

ZNB=NSZ. (Fig. 1.) Thus NX-NS; and the point of tangency, X, of the path of any particle to the line ZR, lies in a parabola, which has N for its focus, and V for its vertex.

As the orbits of all the particles are tangent to this parabola, the paraboloid generated by revolving it about its axis will be, approximately, the bounding surface of the head of the comet. To ascertain the form of the orbit traced, on the present supposition, by any particle, resume the value of t, as given by equ. (5); also take the value of t given by equation x=k cos a. t2. We thus get

√2pr

k sin a
2pr

k sin a

=

Squaring, k2 sin 2a

pr

SECOND SERIES, VOL. XXVII, No. 79.--JAN., 1859.

Transposing coordinates to point X, we have, x=

pr -z'; substituting and reducing,

and z=
k sin a

tance XY =

parameter
4

pr cos a
k sin 2a

-x',

cos 28, draw YK perpendicular to the diameter; the vertex will be on this line. Through the point n where this line crosses the tangent XZ, draw nm perpendicu

lar to XZ, and with a radius =

parameter
4

describe a circle

around X as a centre; the point F in which this circle cuts nm is the focus. Drawing FK parallel to XP, we have the axis and the vertex. The parameter of the axis= X cot a sinẞ. sin 2ẞ= X cot a sin 38=Z cot a sin 38 (by equs. (2) and (7)).

In this investigation it is virtually supposed that the nucleus moves along the line NL, perpendicular to NZ, at the same rate that the particle recedes from NZ. This may seem to be a rough approximation except in the case of a large angle of emission; but, with a certain restriction soon to be stated, the results obtained are in fact a close approximation to the truth. The reason is that the repulsive force of the nucleus produces the greater part of its effect while the particle is in its vicinity, and at a distance from it, where the change of direction and the difference between NZ and NX has become considerable, it is exceedingly small in comparison with the sun's action. To test the accuracy of our results, let us investigate the distance that