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Updated: May 16, 2025
Hopkinson communicated to this institution the results of some very careful experiments made on a Siemens machine. He measured the force expended by means of a registering dynamometer, and obtained very high coefficients of efficiency, amounting to nearly 90 per cent. M. Hagenbach also obtained from one machine a result only a little less than unity.
The laws of numbers in physics are often only laws of proportion. We transform them into laws of equation, because we introduce numerical coefficients and choose the units on which they depend so as to simplify as much as possible the formulas most in use.
If now we calculate the binomium of Newton for different values of the exponent, the sum of the coefficients is doubled for each higher unit of the exponent, and at the same time the extreme limit of the curve is extended one step farther. Hence it is possible to calculate a relation between the value of the extreme and the number of cases required.
The researches on the secular variations of the eccentricities and inclinations of the planetary orbits depend upon the solution of an algebraic equation equal in degree to the number of planets whose mutual action is considered, and the coefficients of which involve the values of the masses of those bodies.
Remember how small-pox often disappears from a community in spite of its contagious character, and the necessary exposure of many persons to those suffering from it; in both diseases contagion is only one of the coefficients of the disease. I have already spoken of the possibility that Dr. Meigs may have been the medium of transfer of puerperal fever in some of the cases he has briefly catalogued.
All that you mean to say, therefore, is that the present state of the system is defined by equations into which differential coefficients enter, such as ds|dt, dv|dt, that is to say, at bottom, present velocities and present accelerations. You are therefore really speaking only of the present a present, it is true, considered along with its tendency.
The new dispensation offers two new factors bearing on this businesslike need of a sagacious sabotage, or rather it brings a change of coefficients in two factors already familiar in business management: a greater need, for gainful business, of resorting to such limitation of traffic; and a greater facility of ways and means for enforcing the needed restriction.
Thus the calculation of the J's the coefficients of impetus, as I will call them does not involve anything very revolutionary in the mathematical knowledge of physicists. We now return to the path of the attracted particle.
By means of the Arithmetical Triangle, an invention equally ingenious and original, he succeeded in solving a number of theorems which it would have been difficult to demonstrate in any other way, and in finding the coefficients of different terms of a binomial raised to an even and positive power.
The loss of pressure in question can only be calculated conveniently on the hypothesis that it is very small, and the general formula, For this purpose the binomial formula, au + bu squared, or the simple formula, b1 u squared, is generally adopted; a b and b1 being coefficients deduced from experiment.
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