observed in nature are as sharp as those here represented, which are intentionally exaggerated, while in all which has just preceded, by an equally intentional exaggeration of the normal action, the wind-pulsations have been supposed to alternate with absolute calm. This being understood, it is scarcely necessary to point out that if the calm is not absolute, but if there are simply frequent successive winds or pulsations of wind of considerably differing velocity (such as the anemometer observations show are realized in nature), the same general effect will obtain*, though we are not entitled to assume from any demonstration thus far given that the total advance will be necessarily greater than that of the whole distance the mean wind has travelled. It may also be observed that the actual actions of the soaring bird may be, and doubtless are, more complex in detail than those of this diagram, while yet in their entirety depending on the principles it sets forth. The theoretical possibility at least will now, it is hoped, be granted, not only of the body's rising indefinitely, or of its descending in the interval of calm to a higher level C than it rose from at A, but of its advancing against the calm or light wind through a distance BC, greater than that of AB, and so on. The writer, however, repeats that he has reason to suppose, from the data obtained by him, that this is not only a theoretical possibility, but a mechanical probability under the conditions stated, although he does not here offer a quantitative demonstration of the fact, other than by pointing to the movements of the soaring bird and inviting their reconsideration in the light of the preceding statements. The bird, by some tactile sensibility to the pressure and direction of the air, is able, in nautical phrase, to "see the wind" and to time its movements so that, without any reference to its height from the ground, it reaches the lowest portion of its descent near the end of the more rapid windpulsation; but the writer believes that to cause these adaptive * The rotation of the body about a vertical axis so as to change the aspect of the inclination, as in the first figure, may be illustrated by the well-known habit of many soaring birds, of moving in small closed curves or spirals, but it may also be observed, in view of the fact that even in intervals of relative calm during which the body descends, there is always some wind, that in making the descents, if the body, animate or inanimate, maintain its direct advance, this wind tends to strike on the upper side of the plane or pinion. Mr. G. E. Curtis offers the suggestion that the soaring bird avoids such a position when possible, and therefore turns at right angles to, or with, the wind, and that this may be an additional reason for its well-known habit of moving in spirals. † Mouillard. changes in an otherwise inert body, with what might almost be called instinctive readiness and rapidity, does not really demand intelligence or even instinct, but that the future aerodrome may be furnished with a substitute for instinct, in what may perhaps allowably be called a mechanical brain, which yet need not, in his opinion, be intricate in its character. His reasons for this statement, which is not made lightly, must, however, be reserved for another time. It is hardly necessary to point out that the nearly inert body in question may also be a human body, guided both by instinct and intelligence, and that there may thus be a sense in which human flight may be possible, although flight depending wholly upon the action of human muscles be for ever impossible. Let me resume the leading points of the present memoir in the statement that it has been shown : (1) That the wind is not even an approximately uniform moving mass of air, but consists of a succession of very brief pulsations of varying amplitude, and that, relatively to the mean movement of the wind, these are of varying direction. (2) That it is pointed out that hence there is a potentiality of "internal work " in the wind, and probably of a very great amount. (3) That it involves no contradiction of known principles to declare that an inclined plane, or suitably curved surface, heavier than the air, freely immersed in, and moving with the velocity of the mean wind, can, if the wind-pulsations here described are of sufficient amplitude and frequency, be sustained or even raised indefinitely without expenditure of internal energy, other than that which is involved in changing the aspect of its inclination at each pulsation. (4) That since (A) such a surface, having also power to change its inclination, must gain energy through falling during the slower, and expend energy by rising during the higher, velocities; and that (B) since it has been shown that there is no contradiction of known mechanical laws in assuming that the surface may be sustained or may continue to rise indefinitely, the mechanical possibility of some advance against the direction of the wind follows immediately from this capacity of rising. It is further seen that it is at least possible that this advance against the wind may not only be attained relatively to the position of a body moving with the speed of the mean wind, but absolutely, and with reference to a fixed point in space. (5) The statement is made that this is not only mechanically possible, but that, in the writer's opinion, it is realizable in practice. Finally, these observations and deductions have, it seems to me, an important practical application not only as regards a living creature like the soaring bird but still more as regards a mechanically constructed body, whose specific gravity may probably be many hundred or even many thousand times that of the atmosphere. We may suppose such a body to be supplied with fuel and engines, which would be indispensable to sustain it in a calm, and yet which we now see might be ordinarily left entirely inactive, so that the body could supposably remain in the air and even maintain its motion in any direction without expending its energy, except as regards the act of changing the inclination or aspect which it presents to the wind, while the wind blew. The final application of these principles to the art of aerodromics seems then to be, that while it is not likely that the perfected aerodrome will ever be able to dispense altogether with the ability to rely at intervals on some internal source of power, it will not be indispensable that this aerodrome of the future shall, in order to go any distance-even to circumnavigate the globe without alighting,-need to carry a weight of fuel which would enable it to perform this journey under conditions analogous to those of a steamship, but that the fuel and weight need only be such as to enable it to take care of itself in exceptional moments of calm. Smithsonian Institution, THE XLII. A new Electrical Theorem. By THOMAS H. BLAKESLEY, M.A.* HE very short paper which I shall read to the Society contains the account of a Theorem which, though admitting of easy proof, appears, so far as my inquiries have gone, to have hitherto escaped notice. In order to state the matter briefly, it will be well to adopt the following definition : If in any system of conductors, however reticulated, two or more modes of disposition of sources of electromotive force produce in every part of the network the same current, such systems of disposition are called equivalent systems. Then the theorem is as follows:-In any system of conductors, possessing seats of electromotive force at any number of points, if any of these sources be supposed to move continuously along the various bars of the conducting system, Communicated by the Physical Society: read February 23, 1894. and, where a point of junction is encountered, each to become a seat of the same electromotive force in each of the newly encountered bars (of course leaving the resistance of the source behind), then the disposition at any moment is equivalent to that at any other moment, and therefore to the original disposition. [Of course the direction of the electromotive force must be carefully maintained the same: if it is towards the knot before crossing it, it must be away from the knot after passing it.] The proof need only refer to the passing of a knot point, for no one will doubt that if the sources only move in an unbranching portion of the conductor the currents in different parts of the system will remain the same. A Let therefore the source e approach the point A at which its path splits into n other ways. In each of the n bars suppose a source e inserted as directed, then these n alone must be equivalent to the single source before reaching A; for if the n sources are reversed, the current due to these sources in every portion of the system is reduced to zero. The reversed n sources would therefore alone produce currents in the system equal numerically, but opposite in direction, to those produced by the single source. Hence it follows that the n sources (not reversed) will produce the same current as the single source. The principle of the superposition of currents enables us to apply this result to each source of the system, and therefore to prove the truth of the theorem in its complete generality. În equivalent systems, since the current in every part remains the same, the total power expended remains the same; and equivalent systems might have been defined as those which produce the same expenditure of power in each part; and therefore the total power of the sources remains the same. From the point of view of Kirchhoff's theorem Σe=Σ.rc for any closed path in a network, the above general theorem may seem to some minds even plainer and more easily proved than on the method of demonstration which I have employed. For it is plain from the method of derivation, that if a seat of electromotive force exists in any closed Kirchhoff path it can never leave it; and if in the movement of the sources one of them approaches the closed path under consideration, at the encounter it becomes in that path two equal sources acting in opposite directions. If, therefore, Σe remains the same for any path and r remains the same for every part, then obviously c must remain the same for any portion of that path, and therefore for every part of the network. The following propositions flow immediately from the main proposition : (1) If a closed continuous surface contains any region of any network, and some bar which cuts the surface contains, or by derivation as above can be made to contain, the seat of an electromotive force, then that source can be done away with without disturbing the currents in any portion of the system provided that in the other bars which cut the surface sources of electromotive force be inserted of equal value but of opposite direction as regards inside and outside of the surface; for it is clear that such sources would result from the migration of sources in one direction. (2) If two systems of electromotive forces are equivalent, one may be derived from the other. For if system A is equi valent to system B, and we suppose A 2 to represent a distribution identical with A as regards positions, but of half the Α B electromotive force in any case, then + is equivalent to 2 2 A or B alone. Now if any Kirchhoff path containing a A B source from does not also contain a source from then 2' 2 Kirchhoff's law would be outraged; for the sum of the electromotive forces in that path would be only half what they are from A alone, whereas the currents and resistances remain the same. Hence for every Kirchhoff path there must be equal sources from each system. Either system may now have its elements moved up to those of the other system; any resulting side branchings will be the same (though differing in sign) whether derived from or from A В 2' and must necessarily produce no effect by themselves, because if B B 2' 2 we consider to be reversed, the whole effect of and A now in the same bars, will be zero, |