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Updated: May 16, 2025
Newton discovered that the law of chance can be expressed by very simple mathematical calculations. Without going into details, we may at once state that these calculations are based upon his binomium. For this construction the coefficients are used as ordinates, the length of which is to be made proportionate to their value.
This is wrong. Some people prefer leisure, or freedom, or predictability to expected money. In other words, people have many utility functions running simultaneously or, at best, one utility function with many variables and coefficients.
Q. What will be the average difference between the nominal and actual powers in the several classes of vessels you have mentioned and the respective coefficients when corrected for the actual power?
But we may suppose that they feel what is done to them, and desire and will what they themselves do. Even intelligence, which we must deny to them in the present, may be attributed to them in the past. Before man existed, the earth, at that time an intelligent being, may have exerted "its physico-chemical activity so as to improve the astronomical order by changing its principal coefficients.
From the circumstance of the third of the new coefficients being greater than the fourth, it appears that the superior result in the fourth set of experiments arose altogether from a greater excess of the actual over the nominal power. Q. These experiments, you have already stated, were all made on paddle vessels. Have similar coefficients of performance been obtained in the case of screw vessels?
A. In the first class of vessels experimented upon, the actual power was about 1.6 times greater than the nominal power; in the second class, 1.67 times greater; in the third class, 1.7 times greater; and in the fourth, 1.96 times greater; while in such vessels as the Red Rover and City of Canterbury, it is 2.65 times greater; so that if we adopt the actual instead of the nominal power in fixing the coefficients, we shall have 554 as the first coefficient, 694 as the second, 832 for the third, and 806 for the fourth, instead of 925, 1160, 1430, and 1580 as previously specified; while for such vessels as the Red Rover, Herne, Queen, and Prince of Wales, we shall have 962 instead of 2550.
These smaller coefficients, then, express the relative merits of the different vessels without reference to any difference of efficacy in the engines, and it appears preferable, with such a variable excess of the actual over the nominal power, to employ them instead of those first referred to.
He has represented the results of his experiments by the binomial formula, au + bu squared, and gives values for the coefficients a and b, which diminish with an increase in diameter, but would indicate greater losses of pressure than D'Aubuisson's formula.
The values, however, which are to be given to these coefficients are not constant, for they vary with the diameter of the pipe, and in particular, contrary to formerly received ideas, they vary according to its internal surface.
A. The coefficients of a greater number of screw vessels have been obtained and recorded, but it would occupy too much time to enumerate them here. The coefficient of performance of the Fairy is 464.8; of the Rattler 676.8; and of the Frankfort 792.3. This coefficient, however, refers to nautical and not to statute miles.
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