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General Scholium and Remarks.
d d - 1. The value ; – 1, = *F, is called the compression of the terrestrial spheroid, and it manifestly becomes known . d . when the ratio F is determined. But the measurements of
philosophers, however carefully conducted, furnish resulting compressions, in which the discrepancies are much greater than might be wished. General Roy has recorded several of these in the Phil. Trans. vol. 77, and later measurers have deduced others. Thus, the degree measured at the equator by Bouguer, compared with that of France measured by Mechain and Delambre, gives for the compression ; also d = 3271208 toises, c = 3261443 toises, d-c = 9765 toises. General Roy's sixth spheroid, from the degrees at the equa
tor and in latitude 45°, gives so Mr. Dalby makes d = 3489932 fathoms, c = 3473656, Gen. Mudge d = 3491420, c = 3468007, or 7935 and 7882 miles. The degree mea
sured at Quito, compared with that measured in Lapland by Swanberg, gives compression = so Swanberg's observations, compared with Bouguer's, give so Swanberg's compared with the degree of Delambre and Mechain so # A minimum of errors in Lapland, France, and Peru gives so Laplace,
Compared with Major Lambton’s degree
from the lunar motions, finds compression - so From the theory of gravity as applied to the latest observations of Burg, Maskelyne, &c. 309 05" From the variation of the pendulum
- - - - l - in different latitudes ##, Dr. Robison, assuming the va. riation of gravity at w makes the compression # The most accurately computed results from Capt. Sabine's expe
riments on the pendulum in different latitudes, give ;"
- - l l Others give results varying from H. i to #: but far the
greater number of observations differ but little from 30.’
* See Ivory in Phil. Mag. July 1826.
which the computation from the phenomena of the precession of the equinoxes and the nutation of the earth's axis, gives for the maximum limit of the compression.
2. From the various results of careful admeasurements it happens, as Gen. Roy has remarked, “that philosophers are not yet agreed in opinion with regard to the exact figure of the earth; some contending that it has no regular figure, that is, not such as would be generated by the revolution of a curve around its axis. Others have supposed it to be an ellipsoid; regular, if both polar sides should have the same degree of flatness; but irregular if one should be flatter than the other. And lastly, some suppose it to be a spheroid dif. fering from the ellipsoid, but yet such as would be formed by the revolution of a curve around its axis.” According to the theory of gravity, however, the earth must of necessity have its axes approaching nearly to either the ratio of 1 to 680 or of 303 to 304; and as the former ratio obviously does not obtain, the figure of the earth must be such as to correspond nearly with the latter ratio.
3. Besides the method above described, others have been proposed for determining the figure of the earth, by measure. ment. Thus, that figure might be ascertained by the measurement of a degree in two parallels of latitude; but not so accurately as by meridional arcs, 1st. Because, when the distance of the two stations, in the same parallel, is measured, the celestial arc is not that of a parallel circle, but is nearly the arc of a great circle, and always exceeds the arc that corresponds truly with the terrestrial arc. 2dly. The interval of the meridian's passing through the two stations must be determined by a time-keeper, a very small error in the going of which will produce a very considerable error in the computation. Other methods which have been proposed, are, by comparing a degree of the meridian in any latitude, with a degree of the curve perpendicular to the meridian in the same latitude; by comparing the measures of degrees of the curves perpendicular to the meridian in different latitudes; and b comparing an arc of a meridian with an arc of the parallel of 1atitude that crosses it. The theorems connected with these and some other methods are investigated by Professor Playfair in the Edinburgh Transactions, vol. v. to which, together with the books mentioned at the end of the 1st section of this chapter, the reader is referred for much useful information on this highly interesting subject.
Having thus solved the chief problems connected with Trigonometrical Surveying, the student is now presented with the following examples by way of exercise.
Vol. II. 14
Ex. 1. The angle subtended by two distant objects at a third object is 66°3039"; one of those objects appeared under an elevation of 25'47", the other under a depression of 1". Required the reduced horizontal angle. Ans. 66 30.36%".
Er. 2. Going along a straight and horizontal road which passed by a tower, I wished to find its height, and for this purpose measured two equal distances each of 84 feet, and at the extremities of those distances took three angles of elevation of the top of the tower, viz. 36-50, 2124', and 14°. What is the height of the tower Ans. 53-96 feet.
Ea. 3. Investigate General Roy's rule for the spherical excess, given in the scholium to prob. 8.
Er. 4. The three sides of a triangle measured on the earth's surface (and reduced to the level of the sea) are 17, 18, and 10 miles : what is the spherical excess Ans. 1"-096. Er. 5. The base and perpendicular of another triangle are 24 and 15 miles. Required the spherical excess. Ans. 2'21"52;". Ex. 6. In a triangle two sides are 18 and 23 miles, and they include an angle of 50°24'36". What is the spherical excess 1 Ans. 2"-31639.
Er. 7. The length of a base measured at an elevation of 38 feet above the level of the sea is 34286 feet: required the length when reduced to that level ! Ans. 34:285-9379.
Er. S. Given the latitude of a place 48.51'N, the sun's declination 18°30'N, and the sun's apparent altitude at 10' 11" 26 AM, 52°35' ; to find the angle that the vertical on which the sum is, makes with the meridian. Ans. 45°23'2"#. Er. 9. When the sun's longitude is 29°13'43", what is his right ascension ? The obliquity of the ecliptic being 23°27'40". Ans. 27°10'13";. Er. 10. Required the longitude of the sun, when his right ascension and declination are 32°46'52", and 13°13'27"N respectively. See the theorems in the scholium to prob. 12. Er., 11. The right ascension of the star & Ursae majoris is 162°50'34", and the declination 62° 50'N : what are the longitude and latitude 1 The obliquity of the ecliptic being as above. Er. 12. Given the measure of a degree on the meridian in N. lat-49-3, 60833 fathoms, and of another in N. lat. 12-32, 60494 fathoms: to find the ratio of the earth's axes. Er. 13. Demonstrate that, if the earth's figure be that of an oblate spheroid, a degree of the earth's equator is the
first of two mean proportionals between the last and first degrees of latitude.
Er. 14. Demonstrate that the degrees of the terrestrial meridian, in receding from the equator towards the poles, are increased very nearly in the duplicate ratio of the sine of the latitude.
Er. 15. If p be the measure of a degree of a great circle perpendicular to a meridian at a certain point, m that of the corresponding degree on the meridian itself, and d the length of a degree on an oblique arc, that arc making an angle a with the meridian, then is d = −."----. Required a
P + \m-p) sin” a
demonstration of this theorem.
ON THE NATURE AND SOLUTION OF EQUA. TIONS IN GENERAL.
1. IN order to investigate the general properties of the higher equations, let there be assumed between an unknown quantity r, and given quantities a, b, c, d, an equation constituted of the continued product of uniform factors: thus (r—a) × (4 —b) × (r-c) x (a — d) = 0. This, by performing the multiplications, and arranging the final product according to the powers or dimensions of r, becomes a"—a Ya" + abi) a "- abc Y r +abcd = 0. . . . (A)
—b + o —abd
—c + ad —acd
—d ) " + be f —bcd
+ ba |
Now it is obvious that the assemblage of terms which compose the first side of this equation may become equal to nothing in four different ways; namely, by supposing either a = a, or r = b, or a = c, or a = d ; for in either case one or other of the factors x — a, c – b, r — c, a -d, will be equal to nothing, and nothing multiplied by any quantity whatever will give nothing for the product. If any other value e be put for r, then none of the factors e—a, e—b, e—c, e—d, being equal to nothing, their continued product cannot be equal to nothing. There are therefore, in the proposed equation, four roots or values of r ; and that which characterises these roots is, that on substituting each of them successively instead of z, the aggregate of the terms of the equation vanishes, by the opposition of the signs -- and —.
‘I he preceding equation is only of the fourth power or degree; but it is manifest that the above remark applies to equations of higher or lower dimensions: viz. that in general an equation of any degree whatever has as many roots as there are units in the exponent of the highest power of the unknown quantity, and that each root has the property of rendering, by its substitution in place of the unknown quantity, the aggregate of all the terms of the equation equal to nothing. It must be observed that we cannot have all at once r = a, r = b, c = c, &c. for the roots of the equation ; but that the particular equations a — a = 0, x – b = 0, r – c = 0, &c. obtain only in a disjunctive sense. They exist as factors in the same equation, because algebra gives, by one and the same formula, not only the solution of the particular problem from which that formula may have originated, but also the solution of all problems which have similar conditions. The different roots of the equation satisfy the respective conditions; and those roots may differ from one another, by their quantity, and by their mode of existence. It is true, we say frequently that the roots of an equation are r = a, a = b, a = c, &c. as though those values of x existed conjunctively; but this manner of speaking is an abbreviation, which it is necessary to understand in the sense explained above. 2. In the equation A, all the roots are positive ; but if the factors which constitute the equation had been z + a, r + b, a + c, z + d, the roots would have been negative or subtractive. Thus a"+a ) a "+ah ) a "+abc ) or + abcd = 0. . . . (B) + b ( --ac | +abd + c ( -Had +acd +d) - bc ( --bcd +bd +cal J has negative roots, those roots being w = — a, z = — b, r = — c, r = — d. and here again we are apt to apply them disjunctively. 3. Some equations have their roots in part positive, in part negative. Such is the following:
negative root, viz. z = - c : the equation being constituted of the continued product of the three factors, a - a = 0, & — b = 0, x + c = 0.