## Theory of the Motion of the Heavenly Bodies MOving About the Sun in Conic Sections |

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aberration according accuracy aphelion approximate value ascending node auxiliary angle auxiliary quantity axis calculation celestial sphere circle Clog coefficients comets computed conic sections conveniently corrected value corresponding cos2 cotan deduced denote derived determined difference differential distance earth eccentric anomaly eccentricity ecliptic elements ellipse equal equation error evident example expressed former fundamental plane geocentric place given heavenly body heliocentric motion heliocentric place Hence hyperbola hyperbolic motion inclination intersection intervals known lastly latter Log diff log Q logarithm LogB logr longitude and latitude longitude in orbit manner mean anomaly method of article moreover necessary negative nutation observed places obtained parabola parallax perihelion perturbations plane preceding article precision problem radii radius vector readily reduced referred regarded remaining right ascension right line roots sin2 sin4 sine solution substituted taken tan2 tion true anomaly unknown quantities whence Wherefore wholly

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Page 260 - Therefore., tliat will be the most probable system of values of the unknown quantities p, q, r, s, etc., in which the sum of the squares of the differences between the observed and computed values of the functions V, V, V", etc. is a minimum, if the same degree of accuracy is to be presumed in all the observations.

Page 249 - ... all computations made concerning concrete phenomena must be to approximate, as nearly as practicable, to the truth. But this can be accomplished in no other way than by a suitable combination of more observations than the number absolutely requisite for the determination of the unknown quantities. This problem can only be properly undertaken when an approximate knowledge of the orbit has been already attained, which is afterwards to be corrected so as to satisfy all the observations in the most...

Page 260 - V", etc. is a minimum, if the same degree of accuracy is to be presumed in all the observations. This principle, which promises to be of most frequent use in all applications of the mathematics to natural philosophy, must, everywhere, be considered an axiom with the same propriety as the arithmetical mean of several observed values of the same quantity is adopted as the most probable value.

Page 249 - If the astronomical observations and other quantities, on which the computations of orbits is based, were absolutely correct, the elements also, whether deduced from three or four observations, would be strictly accurate (so far indeed as the motion is supposed to take place exactly according to the laws of Kepler), and, therefore, if other observations were used, they might be confirmed but not corrected. But since all our measurements and observations are nothing more than approximations to the...

Page xv - I ever have found a more seasonable opportunity to test the practical value of my conceptions, than now in employing them for the determination of the orbit of the planet Ceres, which during these forty-one days had described a geocentric arc of only three degrees, and after the lapse of a year must be looked for in a region of the heavens very remote from that in which it was last seen? This first application of the method was made in the month of October, 1801, and the first clear night, when the...

Page 271 - And when it can be assumed that these are so small that their squares and products may be neglected, the...

Page 260 - the most probable value of the unknown quantities will be that in which the sum of the squares of the differences between the actually observed and the computed values multiplied by numbers that measure the degree of precision is a minimum".

Page 260 - ... a double error can be committed in the former system with the same facility as a single error in the latter, in which case, according to the common way of speaking, a double degree of precision is attributed to the latter observations.”” The fact of the matter is, however, that: “. . . different fields have particularly favorite ways of expressing precision. Most of these measures are multiples of the standard deviation; it is not always clear which multiple is meant. . “Some consider...

Page 259 - ... probability of errors exceeding those limits ought always to be zero. while our formula always gives some value. However, this defect, which every analytical function must, from its nature, labor under, is of no importance in practice, because the value of our function decreases so rapidly, when liA has acquired a considerable magnitude, that it can safely be considered as vanishing.

Page 258 - ... base, the common principle, the excellence of which is generally acknowledged, depends. It has been customary certainly to regard as an axiom the hypothesis that if any quantity has been determined by several direct observations, made under the same circumstances and with equal care, the arithmetical mean of the observed values affords the most probable value, if not rigorously, yet very nearly at least, so that it is always most safe to adhere to it. By putting, therefore, V = V = V