Power series

“ In this series the terms of order n are sums, with real coefficients, of the various integral polynomials of dimension n which satisfy the equation ∂2ψ/∂x2 + ∂2ψ/∂y2; the series is thus the real part of a power series in z, and is capable of differentiation and integration within its region of convergence. ”

“ This converges for values of between and 2 and is the expansion of in the vicinity of , the remainder converging to zero. When a function of the sum of two numbers and is given, say , it is frequently desirable to expand the function into a power series in one of them, say . ”

“ Perhaps the simplest possible power series is ez=exp (z) =1 + z2/2! + z3/3! + . . . of which the radius of convergence is infinite. By multiplication we have exp (z) .exp (z1) =exp (z+z1) . In particular when x, y are real, and z=x+iy, exp (z) =exp (x) exp (iy) . Now the functions
U0=sin y, V0=1 − cos y, U1=y − sin y,
V1=1/2y 2 − 1 + cos y, U2=1/6y 3 − y + sin y, V2=1/24y 4 − 1/2y 2 + 1 − cos y, . ”

“ Since variation of the field strength does not materially affect the voltage, by adjusting the resistance in series with the magnetic circuit, the strength of the field can be changed and the power factor kept 1 or nearly 1 as different loads are thrown on and off the rotary. ”

“ As to the sudden industrial progress which has been achieved during our own century, and which is usually ascribed to the triumph of individualism and competition, it certainly has a much deeper origin than that. Once the great discoveries of the fifteenth century were made, especially that of the pressure of the atmosphere, supported by a series of advances in natural philosophy--and they were made under the medieval city organization,--once these discoveries were made, the invention of the steam-motor, and all the revolution which the conquest of a new power implied, had necessarily to follow. ”

“ It would be our interest therefore to keep far away from such a stream, lest, engaged on temporal things, we should neglect eternity. How can a man keep for ever anything here, be his love for it never so passionate? Which of life’s most cherished objects endures always? What flower of prime? What gift of strength and beauty? What wealth, or fame, or power? They all have their transient bloom, and then melt away into their opposites. Who can continue in life’s prime? Whose strength lasts for ever? Has not Nature made the bloom of beauty even more short-lived than the shows of spring? ”

“ When a complete revolution is accomplished, the conditions existing at the start are reestablished and the same action is repeated for the next and all subsequent revolutions, and in general, it will now be seen that every revolution of the armature of the generator produces a corresponding shifting of the poles or lines of force around the ring. ”

“ I am aware that this view is not that of a practical engineer, but neither is it that of an illusionist, for it is certain, that power transmission, which at present is merely a stimulus to enterprise, will some day be a dire necessity. ”

“ The individual worlds are in the charge of Master Physical Controllers and receive the encircuited lines of power dispatched by the power center of their system. Only those spheres of the most extraordinary energy relationships have power centers of the seventh order acting as universe balance wheels or energy governors. In every phase of activity these power centers are the full equals of those who function on the higher units of control, but not one space body in a million harbors such a living power organization. ”

“ Yet the outer rim of the big generators is spinning at the pace of a hundred thousand miles an hour; the dazzling clean switch-board, with its little handles and levers, is the seat of empire over more power than the strength of a million disciplined, unquestioning men. ”

“ But among all these many departments of research, these many branches of industry, new and old, which are being rapidly expanded, there is one dominating all others in importance — one which is of the greatest significance for the comfort and welfare, not to say for the existence, of mankind, and that is the electrical transmission of power. ”

“ During such a cycle, the work done by the man on the weight is equal to the work done by the weight on the man and no useful external work has been accomplished.
Why the Power Factor is equal to Cos φ.—In the preceding figures showing power curves for various phase relations between current and pressure, the curves show the instantaneous values of the fluctuating power, but what is of more importance, is to determine the average power developed. ”

“ This is the age of concentration or specialization of energy. The problem of the day is to get ten-horse power out of an engine that shall occupy the space of a one-horse power engine, and no more. Just so society demands a ten-man power out of one individual. ”

“ When linked together, machines of struggle are able to confront and challenge the vectoral regime as a real force, collectively armed against the territorializing effects of vectoralist capital. Circuits of counter-power provide the conditions and capacity for transformative constitutive action. Such circuits are but one moment of the potential power of the creative multitude as organised and effective transformative agents. ”

“ These systems will continue to be modified, until this tendency shall have opened such channels for itself, as will permit free and untrammeled action; until these channels shall have encircled the world, and its utmost limits shall have been attracted within the realm of its positive flow and negative reaction, and until the commanding magnetic influence that shall proceed from its central seat of power shall reach all subjects and find in their general heart an answering response of fidelity and confidence. ”

“ We may omit, for the time, from our notice, the before-mentioned possibility of a knowledge of the world that would reduce it to a single serial order of the general type of the Well-Ordered Series of whole numbers, where every term has one coming next after it. If we abstract from that possibility, we are left to the conception that Between any two facts there are to be found various series of intermediaries of the type now defined. The world thus regarded will consist for us of all these interwoven series, and will constitute a single System. ”

“ It is not alone in the greater operations of the engineer or the manufacturer, that those vast powers which man has called into action, in availing himself of the agency of steam, are fully developed. Wherever the individual operation demanding little force for its own performance is to be multiplied in almost endless repetition, commensurate power is required. It is the same 'giant arm' which twists 'the largest cable', that spins from the cotton plant an 'almost gossamer thread'. ”

“ Morey could read the utter despair of the man.
"What is the difficulty?" asked Morey eagerly.
"Power, lack of power. Our cities are going without power, while every electric generator on the planet is pouring its output into the accumulators that work these damnable, hopeless things. Invincible with power—helpless without."
"Ah!" Morey's face shone with delight—invincible weapon—with power. And the Ancient Mariner could generate unthinkable power. ”

“ The governor, and other methods of regulating the motion of machinery which have been just described, are adapted principally to cases in which the proportion of the resistance to the load is subject to certain fluctuations or gradual changes, or at least to cases in which the resistance is not at any time entirely withdrawn, nor the energy of the power actually suspended.236 Circumstances, however, frequently occur in which, while the power remains in full activity, the resistance is at intervals suddenly removed and as suddenly again returns. ”

“ These beings receive the incoming lines of power from the third-order centers of Uversa and relay the down-stepped and modified circuits to the power centers of our constellations and systems. These power centers, in association, function to produce the living system of control and equalization which operates to maintain the balance and distribution of otherwise fluctuating and variable energies. ”

“ It will be remembered, that at the meeting of the British Association at Southampton in 1882, the late Sir W. Siemens proposed that the unit of power should be the watt, and that the watt, which was derived from the C.G.S. system of absolute units, should in future, among electricians, be the unit of power. ”

“ It is obviously clumsy to create power in the first instance, convert power into electricity (I use popular language) , and then convert electricity back again into power. Much more hopeful than any idea of developing that method would be the conception of new ways of creating and applying motive-power directly. But, almost certainly, electricity, obtained in some new way, will do the work of the world for many generations yet—until, in fact, we devise or discover something more convenient. ”

“ Young Halsey, dull white, simply sat staring at him as he went on.
"We all know what the old ideas of fuel and power are—they're obsolete. Electricity is the power of the future, the power of to-day. Speed, speed, speed is what we want. Power, power, power is what every industry needs, as well as what every man craves. ”

“ It is thought that our modern life is more complex than that of the eighteenth century, because the machinery by which it is carried on is expanded. Transportation, newspapers, corporations, oceans of books and magazines, foreign cables, have changed the forms by which power is transmitted. ”

“ One plan may use up an indefinite number of separate stores of energy. Even in an individual the realisation of a plan exhausts the powers of millions of organic cells. A military campaign illustrates the relation of plan to power. The design may have been formed by one man, and then communicated wholly or partially to a hundred thousand, and the energies of these may be devoted to its realisation. ”

“ One can hardly imagine a modern factory in full working being equipped with only one engine, which might possibly have an excess of power necessary to drive all the machines. In the case of breakdown, which sometimes happens in the best supervised factories, it would be small consolation to know that this excess of power was present theoretically. ”

“ The value of the latter is important inasmuch as the valuable features of a series generator assert themselves only when the machine is supplying a greater number of amperes than that of the critical current, for if the series generator be worked along that part M A of the curve to the right of the point of saturation it becomes nearly self-regulating as regards current, because as the current increases the voltage drops. ”

“ These conditions have naturally tended toward the development of a great variety of systems, employing more or less complicated circuits and apparatus, and although alternating current practice is still much less definite than direct current work, certain polyphase systems are now being generally accepted as representing the highest standards of power generation, transmission and distribution. ”

“ Franklin's experiments in electricity were followed at once by the common lightning-rod, but a century passed before the electrical power was utilized, and made subservient, in some degree, to the control of men. ”

“ Right at the back of this house is a wire, a power-circuit, alive with more than two hundred times that voltage, with power in plenty—a little condensed Niagara of power—asking to be taken off and made use of!”
“But what use?”
“I can capture and tame and control that power, Mame. I can make it my slave and carry it along with me, almost in my pocket, on a mere thread of wire. I can make it a living, iron-eating otter, with a dozen fangs—in the shape of quarter-inch drills, gnawing and biting and eating through that armor-plate door about the same as a rat would gnaw through a wooden lath. ”

“ In the present day, the power really exerted is equal to the force of a hundred millions of men at least. But the power of the uneducated unit is still the physical force of one man, the balance being exerted by men who understand the principles of mechanics and of chemistry, and who superintend the machine power evolved thereby. Thus the power originated by the few, and superintended by a fraction of society, is seventeen times greater than the strength of all our workers, and is hourly increasing. ”

“ I remembered, of course, that the World's Series had been fixed in 1919, but if I had thought of it at all I would have thought of it as a thing that merely happened, the end of some inevitable chain. It never occurred to me that one man could start to play with the faith of fifty million people—with the single-mindedness of a burglar blowing a safe. ”

“ The people of this far away orb have greatly surpassed us in controlling and utilizing the three distinct forces which are quite similar to electricity, and these are the wizard forces that furnish the power used to drive the motors and engines, not only of the floating cities, but also of the fixed abodes. ”

“ If any mode could be discovered of transmitting power, without much loss from friction, to considerable distances, and at the same time of registering the quantity made use of at any particular point, a considerable change would probably take place in many departments of the present system of manufacturing. A few central engines to produce power, might then be erected in our great towns, and each workman, hiring a quantity of power sufficient for his purpose, might have it conveyed into his own house ”

“ It is possible, then, for time to supply a deficiency of power. I can make such a substitution applicable in two ways, by dividing the resistance, or by using a machine as a lever, which, when everything about it is considered, is nothing more or less than a device by means of which we replace power with time. ”