to the luminosity of the atmospheric elements will, in its passage through the non-luminous lower strata of the atmosphere, be very greatly diminished, owing to absorption; whilst the light due to any substance not present in the lower strata will be almost wholly transmitted to the eye. Thus the relative intensity of the rays due to the different kinds of matter will, when they reach the eye, be very different from their relative intensity before their passage through the atmosphere*. The absence of a corresponding dark line in the solar spectrum shows, I think, that the quantity of the gaseous matter is very small; and this, taken in conjunction with the brightness of the line in the auroral spectrum, shows that it is confined to the higher parts of the atmosphere. Auroras are known to occur in the lower parts of the atmosphere, in the region of the clouds; and some have been observed so low as to appear against a mountain as a background. It would be interesting to know whether such auroras would exhibit the same spectrum as that observed by M. Ångström. M. Ångström asserts, as a deduction from his observations, that the aurora is not due to electricity. The only other way in which it seems possible to account for luminous matter in the atmosphere is by supposing chemical action to be taking place, and, as all chemical action between the constituents of the atmosphere must have ceased long ago, if ever there was any chemical affinity between them, we must suppose that this chemical action is taking place between the elements of the atmosphere and the newly introduced gaseous matter. I do not, however, see that M. Angström's observation affords any ground for believing that the aurora is not the light due to electrical discharges. A. S. DAVIS. Roundhay Vicarage. * I wish here to point out that the light from a nebula may be due to only one of the different gases composing it; for the light due to the other gas may be wholly absorbed in passing outwards through its non-luminous portions. The presence of lines in the spectrum of a nebula due only to one kind of matter, is therefore no proof that it is not composed of more than one kind of matter. V. On the Solution of Linear Partial Differential Equations of the Second Order involving two Independent Variables. By R. MOON, M.A., Honorary Fellow of Queen's College, Cambridge*. THE HE following method of treating the problem offered to us in the solution of the General Linear Partial Differential Equation of the Second Order, in the cases which are not amenable to Monge's method, may be found to possess both interest and value. Let +S. +T +P +Q+Uz+V, d2z d2z dz dxdy dy da Ꮳ dy (1) where R, S, T,P,Q, U, V are functions of x and y only; and assume ≈≈ A + A¤(u) +A ̧8−1(u) +A28−2(u)+R, &c.,. (2) where u,Ã, A, A1, A2, &c. are functions of x and y only, and Substituting these values in (1) and equating to zero the coefficients of (u), p' (u), p(u), p−1(u), &c., we get d2u dx du dy + (Rd2+S dry +Td2 + d + Qdu)A, dxdy du dx dy 0 = (2R du + s du) dA -(8 d +27 du) da dy) dx + (s dx d2u deu +(Rda + S dedy d2A du dA, da dy da +Qdu). dx dy A1 dA dA dudA2 dy dy Ꭱ dx2 d2A +R. dx2 d2A dy2 +2T da 0=(2R (2R+S + +RA d2A1 dx dy &c. 1+UA,, w1 or 2 may be substituted at pleasure for u in (2). The equations which succeed (4) may be written The integral by Lagrange's method of any one of these equations, as (5), will be of the form A1=L.x(w)+M, where L, M are functions of a and y; x is arbitrary; and ∞ = const. is the integral of = 0=dy-adx. Hence if these values were substituted in (2), and we then put (u) a constant, as we may do, we should have the given equation (1) satisfied by an integral of the form z=F{x, y, x(w)}, where F is a definite, and x an arbitrary function; from which it would inevitably follow that the given equation is soluble by Monge's method, whereas by hypothesis it is not so soluble. We must assume, therefore, the arbitrary function in each case to be zero; so that, in order to find A, A,, &c., we may take the equations where c is constant, it being understood that, if B contains y, the latter variable must be eliminated before the integration by means of the equation w= const., and the constant so introduced must be eliminated after integration by means of the same equation. Hence we shall have A = ce fẞdx It is clear, however, that the substitution of these values would give rise to a term in z =e ̄ßdx, {c$(u)+c1$~1 (u)+c2-2 (u) + &c.}, =e-/Bdx (u); |