which, taking alternately the upper and lower signs, constitutes a pair of equations of condition which must be satisfied when f is susceptible of both the values (9). by Moreover, since if (9) satisfies (7) the latter will be satisfied + V ± √ √ 2 + 4 Rp 3 d ( V + √ √2+4Rp2 +v)}, 2p2 dv 2p + dp (V+ ✓ V2+4Rp2) + 2 V ± √ V2 + 4Rp2 . d (V + √/V2 + 4Rp2). 2p2 Hence if (10) hold, we must have dv V2+4Rp2=0, a conclusion which may be rejected on account of defect in generality*. Therefore when equations (10) hold, (9) must reduce itself to and we shall thus have for ƒ four values in all, viz. those given by the last equation together with those previously found, viz. W1, W2. Now it will be remembered that the original integral which we assumed for (2) was (2 a), or, which is the same thing, It is clear, however, that this last equation may be put under the form fa(xytpv)=X_1{f(xytpv)}, where X-1 denotes an arbitrary function; and if we treat this equation in the same way in which we have treated (12), we shall arrive at precisely the same formulæ for the determination *The grounds upon which I rest this conclusion will hereafter be more distinctly pointed out. off as those at which we have already arrived for the determination of f. Thus the occurrence of two of the four values of which ƒ has been shown to be susceptible is at once accounted for; and if we reflect that almost universally when an equation of the second order is integrable by Monge's method, there are two equations of the first order of the form (2 a) from each of which it is separately derivable, there cannot, I think, be a doubt that the four values above derived for ƒ must be attributable to the values of f and f in equation (12), and to the corresponding functions in the other first integral of (2). Acting from this clue, we shall be justified in assuming, and we shall find it to be the fact, that when the equations of condition (10) are satisfied, (2) will be susceptible of the two following integrals, from each of which separately it is capable of being derived, viz. V+ √ V2 + 4Rp2 . t }• [ 2D (13) If we now recur to the equations of condition (10) or (10 a), or, as they may be written, where dP 0=P2 dp (14) P1 =V+ √V2+4Rp2, P2=V- V2+4Rp2, the form of the equations leads us to conclude that, if neither of the quantities P, P, vanishes (as for instance P1) so as to render dP1 dP. 2 = =0, we shall have dp dv which implies that we have ✔V2+4Rp2=0,-a conclusion which I have aleady pointed out as one to be rejected as deficient in point of generality*. * This may be seen more distinctly as follows. In the case supposed, (13) become whence we get w1= funct. w; and therefore we must have p = funct. of v, since ww2 involve p and v only. Now not only is this conclusion, viz. that = funct. v, clearly defective in point of generality, but, if it were true, we Ρ But if one of the quantities P1, P2 is constant-for instance, if we have 2 V+ √V2+4Rp2=2a, where a is constant, we shall have a value which, it will be found, satisfies both the equations of condition (14). In this case equations (8 a), from which w1, we are to be derived, become a 0=de-{x (v+2)-a}; .. @1=v+, and equations (13) become (15) a ρ If we put u=v+ =, the three equations which determine the circumstances of the motion, i. e. the pressure, density, and velocity, become a2 p= +x(u), ρ might assume p= a function of p only--an assumption which is open to all the objections which in my former paper have been shown to apply to the law of the received theory, viz. p=a2p. results which have been evolved from the general equation (2) without the aid of any subsidiary hypothesis with regard to the nature of the law of pressure, by the simple application to (2) of the ordinary process for the integration of partial differential equations of the second order involving two independent variables*. If the fact of the expression for the pressure containing an arbitrary function, thus causing three arbitrary functions to enter into the complete solution of the problem, occasions surprise, I may observe that it may be shown à priori that such must be the case. 6 New Square, Lincoln's Inn, July 7, 1868. XVI. On the Diammonic Carbonate, or Normal Carbonate of Ammonium. By EDWARD DIVERS, M.D., F.C.S., Lecturer on Natural Philosophy, Charing Cross Hospital. IT T is now nearly thirty years ago since H. Rose wrote, in his classical memoir on the compounds of ammonia with carbonic acid, "the neutral anhydrous carbonate of ammonia cannot be combined with the quantity of water sufficient to convert the ammonia into oxide of ammonium." The carbonates of ammonia have since been reinvestigated by Deville§; and he admits the existence of only two crystalline combinations-the tetrammonic dihydric tricarbonate (or true sesquicarbonate), and the ammonic hydric carbonate (or bicarbonate). Resting on the authority of these distinguished chemists, the text-books tell us that the carbonate of ammonium cannot be isolated. I have now to announce that the normal or diammonic carbonate can be prepared in the simplest manner imaginable. I must say that, with the above assertions before me, I was much pleased to succeed very easily in my efforts to form this body; but I certainly was not astonished, because comparatively recently it has been shown by Dr. Hofmann that two bodies whose place is little beyond the threshhold of chemistry (formic aldehyde and formamide), and which it was said could not be produced, are actually to be prepared in the most ordinary manner. Diammonic carbonate is formed by treating the commercial *The truth of equations (16) may be readily verified, observing that, in virtue of the value there given for p, (2) can be put under the form a2 dp. Dp2 da dux'(u) du † Communicated by the Author. carbonate with a solution of ammonia. Some of this substance dissolves in the ammonia-water; the rest remains, as a skeleton of the original solid, in the condition of a soft mealy semicrystalline mass. This is the normal carbonate. If it be digested for some days in a closed vessel with the ammoniacal liquor after this has had its free ammonia renewed in it by the passage of some ammonia gas through it while kept cool by external means, a remarkable phenomenon is observed. Even when the solid is at first in such quantity as to leave the mixture only semifluid, the whole will gradually, particularly when occasionally agitated, become a solution. Warming the vessel in the early period of the digestion seems to have little effect in hastening this peculiar solution; while cooling it in ice has little effect (if any) in increasing the solidification of the mixture. This interesting occurrence of the slow disappearance of the solid carbonate I shall not in the present paper attempt to explain (though I am quite satisfied that I am able to do so); for my experiments in this direction are, as yet, incomplete. If now to the clear liquid more of the commercial carbonate be added, further solution of it occurs; and now a gentle heat seems favourable to this process. Either on cooling after warming, or by the action of applied cold, fine spicula fill the liquid, almost solidifying it from the commencement of their formation, though, somewhat like the potassic silicofluoride, they do not much diminish the transparency of the whole. These minute crystals, which are identical in composition with the mass left when the commercial carbonate is treated with ammonia, rapidly increase in quantity, but not materially in individual size. They are not grouped in stars or bundles, but diffused uniformly through the liquid. By jolting the vessel, the crystals, if not too numerous, may be shaken together so as to form a shrunken mould, as it were, of the inside of the vessel. Removed from their mother-liquor, or, more correctly, drained from and squeezed free of their mother-liquor, they begin to decompose. But with proper precautions to prevent this decomposition, they can be exhibited in soft masses of minute crystals of brilliant silky lustre. They smell most intensely of ammonia. They dissolve very freely in water, but require about seventy measured parts of ordinary rectified spirit to dissolve them. In strong ammonia solution they dissolve at first very sparingly; but solution continues to go on slowly, as I have above described it doing in the case of the carbonate in the mealy condition. Exposed freely to the air, the salt entirely loses its lustre in a few moments, evolving torrents of ammonia, and becoming at the same time damp from the liberation of water. After a while (very rapidly if disturbed and repeatedly pressed between fresh bibu |