regards as inconsistent with his further results that "complete closure. . . . . reduces transpiration to or nearly to cuticular rate," and "when the stomata are open to their utmost limit the highest rate of transpiration is the maximum of which the leaf is capable." A more satisfactory statement would be that until the stomatal aperture is reduced to a certain very small value the possible rate of transpiration is practically independent of the aperture, and nearly all of the reduction to zero when the stoma closes takes place in the last 2 per cent. of the reduction of aperture. Now Next, consider the effect of wind. Suppose for simplicity that the stomata are arranged in straight rows, the distance between consecutive rows and between consecutive stomata on the same row being b. Then b=1/n. Consider a square column of air of side b. To pass over a stoma it would take a time b/u, and if it were unsaturated at the commencement it would therefore acquire a weight of vapour 2πkрcVob/u, if there were no mutual influence between stomata. suppose the air to have moved forward a distance x, in time x/u. Then the vapour in it will have spread out by diffusion through a radius comparable with 2(ka/u); and if ab is great diffusion parallel to the surface of the leaf and across the wind will have practically ceased, and thus the vapour will occupy half a flat cylinder of radius 2(ka/u)3 ́and thickness b, its centre being of course at the point x. Thus the concentration in it will be of order cVo/x. Further, the number of stomata much affected will be of order 4n2b(kx/u)$=4n(kx/u)3. Similarly, the number of stomata whose influence at this time will have affected the column of air when it has travelled a distance between -b and x+b is 4n(kx/u), and therefore the total concentration produced by them is 4ncV。(k/ux)3. This is then the concentration acquired by a mass of air on account of what happened between times (x+b)/u previously, and the total produced by all times is to be found by summing the series for all such intervals. Put x/br. The total is then 4ncVo(k/ub)Σr, the summation being from r=1 to r=l/b, where l is the distance of the mass from the stoma nearest the margin. When is great this is of the order of 8n2cVo(kl/u). Now b is about 0.05 mm., and thus is usually small compared with the thickness of the layer of rapid shearing. A fortiori a, and hence c, are * See p. 271. smaller. Thus u, being the velocity within the region to which the diffusion from a stoma extends, is much less than the velocity outside. Similarly k is the true coefficient of diffusion, about 0.24 cm.2/sec. The thickness of the layer of shearing being 40/U, it follows that u is of order U2a/40. Thus the quantity just obtained is of order 100n2Vo(kla)/U, which is about 100 Vo. It follows by argument similar to that used in the case of no wind that the earlier stomata saturate the air before the later ones are reached. Thus the total evaporation is not very different from that in the case where the whole surface of the leaf is wet *, and is therefore proportional to 115, where l is now proportional to the linear dimensions of the leaf. This approximation will break down if n'a is much smaller, for then the residual saturation from the earlier stomata may be small compared with Vo, and the rate of evaporation will then be the same as that obtained by summing the results for the individual stomata, each being supposed isolated; this sum is 2n2kpcVA. A similar result may be obtained if u/k is much greater. It must always be noted, however, that this formula can be applied only when the result it gives is less than the rate of evaporation from a wet leaf with the wind blowing over it; otherwise we should again have the absurdity of the evaporation from a part being greater than that from the whole. The best method of determining whether it is better to employ the sum of the possible evaporations from the individual stomata, or to regard the whole surface of the leaf as wet, is probably to calculate the rate of evaporation on both bases and take the smaller of the two results as supplying the correct upper limit to the amount of respiration the leaf can perform. Similar remarks will apply to the possible absorption of substances from the air. It may be remarked that when the number of stomata is so large as to make the problem reduce to that of a wet leaf, the total evaporation is not a function of the number of stomata, but that from any single stoma is inversely proportional to the number. Thus increasing the number diminishes the work thrown on any individual, which may be of some physiological importance. The above investigation concerns only the purely physical * O. Renner, Flora, vol. 100. pp. 451-547 (1910), states on p. 485 that the evaporation from a leaf is the same as that from a water surface side of diffusion. It does not preclude the possibility that a reduction of the stomatal aperture may be associated with a reduction of the rate of evaporation; but it does show that in most cases the cause of such reduction is not the mere extra mechanical obstruction to the passage of water vapour, but must depend on the internal conditions. The importance of these is obvious. For instance, in the problem considered here the air has been supposed saturated when in contact with a stoma and perfectly dry at a great distance. Actually the concentration at a great distance has the finite value Va; and that at a stomatal aperture is probably somewhat less than the saturation concentration. Let it be V1. The latter question is further complicated by the facts that the dissolved substances within the cells must diminish the pressure of saturated vapour; that the leaf is normally at a somewhat higher temperature than its surroundings, so that the pressure of saturated vapour will on this account be greater than that at the temperature of the surroundings; and that owing to internal restrictions to the supply of water to the stoma the vapour-pressure may be reduced. The effect of these changes is that in all the formulæ we must substitute V1-Va for Vo Another complication arises from the fact that the stomata are not usually mere pores with saturated air in their planes; in most cases they are pits sunk in the leafsurface. As long as their number is large this is not likely to produce any great effect on the rate of evaporation, for in exactly the same way as with flat stomata the earlier ones met by the air will partially saturate it, and the air when it meets the later ones will be nearly at the same saturation as that inside them. When the number is small, on the other hand, the formula for evaporation from depressed stomata must be used. For circular cylindrical stomata this gives for the rate of evaporation when the depth is great compared with the radius πnkpa (V1-Va) A XXXI. General Curves for the Velocity of Complete Homogeneous Reactions between Two Substances at Constant Volume. By GEORGE W. TODD, D.Sc.(Birm.), B.A. (Camb.)*. WH [Plate IX.] HEN m molecules of a substance A react with n molecules of a substance B to give one or more resultants, there being no back reaction, the velocity of the reaction is given by dx dt =k. C. CB, where x is the change in the concentration C in time t and k is the velocity constant. If k is known, the changes in concentration for various initial concentrations of the reacting substances can be worked out by integrating the above equation, but the integration often absorbs valuable time. By choosing suitable quantities it is possible to plot curves which will apply generally to all reactions of a similar type. The author has worked out some of these, and puts them on record hoping that they may save much time and labour. Bi-molecular Reaction. If the reaction is bi-molecular of the type A+B+1 or more resultants, the reaction velocity is given by where a, b are the initial concentrations of A, B respectively. The equation may be written Take (i.) initial concentrations equal, i. e. a=b or p=1, The maximum value of X=1. Giving X values up to 1 we get X ...... 0 1 -2 -3 ⚫4 Kt.. .5 -6 7 -8 .9 1-0 0 111 250 428 666 100 150 2:33 4:00 9:00 These are plotted on fig. 1 (p=1), and the curve will apply to any bi-molecular reaction in which the initial concentrations are equal. Take (ii.) one of the substances in excess, say where p>1. Then we have p(1-X)}. b α { log. p-X Kt... 0 036 077 126 183 255 344 386 549 851 1·09 1.42 ∞ These are plotted on fig. 1 (Pl. IX.), and the curves will apply to any bi-molecular reaction of the type A+B+1 or more resultants. Ter-molecular Reaction. Let the reaction be represented by 2A+B 1 or more resultants, then the velocity of reaction is given by Take (i.) equal initial concentrations, i. e. p=1, then |