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4. The first term is the same as that for the area of a plane triangle having the same sides and contained angle: the following terms therefore shew the difference between the areas of the two triangles. Of these terms we may take account of as many as suits our object; but in ordinary cases it will be needless to regard any beyond the two first. Limiting ourselves to these, the difference between the areas of the plane and spherical triangles
, , a b /«2 + 62 ab \ corresponds to an excess represented by -j- sin jg— ——j-cosCJ
or by -^j- sin C j at + 6« — 3 a b cos C. j
5. This expression shews that when C exceeds a right angle (cos C becoming — ) the spherical area must exceed that of the plane triangle. When the two terms within the brackets cancel each other, the two triangles have equal areas; and when the second term exceeds the first, the spherical area will be less than that of the plane triangle.
6. The limits are easily assigned.
7. The sum of a and 6 being given, o> + 6* is a minimum, and a b or
3 a 6 is a maximum when a — b. In this case the triangles are isosceles, and o* -f 6* = 2 o2, and 3 a b = 3 a'; hence the terms within the brackets will cancel each other when cos C = J, or when C = 48° 11' 23". This for equal areas is the maximum of C. With isosceles triangles, if C be less than thiB, the spherical area will be less than that of the plane triangle. 8. Again when cos C is a maximum, C = 0: In this case, o* +b* = 9 a b
i£ „ * , , • , ...,,.43+ V~&~ or 1 + -t — 3 —; the solution of a quadratic will give — = —5 =
2-618 nearly. This is the maximum inequality in the sides so as to have equal areas.
9. In like manner may be found the value of the angle for any given ratio of the sides within these limits; or the angle being given, the ratio of the sides may be found.
10. The following Table shews for given ratios of a and b the value of C giving equal areas:—
11. If the sides were so large in regard to the radius, as that the terms omitted could sensibly affect these results, it would be necessary to take those of the next, and perhaps also of higher orders.
12. To ascertain the actual difference in the areas of the spherical and plane triangles in an extreme case, suppose an equilateral with sides of 11 degrees: the direct formula gives the excess = CI.217; and the difference in the areas of the two triangles will be -3951 square miles, corresponding to an excess of 0r"005245. Onethird of this would be the difference on each angle, and were it ten times as great, it would still be, in Troughton's phrase, a quantity less than what is visible in the telescope.
13. It is almost needless to remark, that the supposed triangle is larger than any which has yet occurred in practice. The great triangle in the French arc, long supposed to be the largest in the world, has an excess of about 39". I have had one observed by day-light on which the excess was
This least side was 80 miles, and the largest 92-6. Such a iangle does not often occur, but even this has only about | of the area of that on which the difference has been shewn to be utterly invisible. 14. But as the greatest difference occurs when C exceeds a right angle, ■ ■ may find the particular angle giving the maximum difference by
making-^ j (a* + t* J sin C — 3 a b sin C cos c| a maximum differentiating, wc have — | (a* + b )cos C — 3 a 4 cos 2 C j dC - •
which scarcely admits of a direct solution, but the indirect solution is very easy.
15. As C is to be greater than a right angle, we may put 90 + X = C
a* + 6« .
is always +, it is plain that X can
not exceed 45°, nor be less than 0. Hence the quantity ——<? will pass
sin X through all its values from 0 to CO every hall'quadrant. By tabulating this, as under, for every degree of X' we shall have by inspection for any ratio of the sides, the approximate angle giving a maximum difference of areas. A nearer approximation may be got by making proportion for the differences between the tabular and actual quantities in the usual way; and by computing another value on each side of the angle Bo found, we may by successive steps bring the approximation as close as we please.
16. By means of this and the former Table, it appears that with equal siili's the angle of maximum difference of areas is somewhat greater than 124°, and by another computation it will be found that the exact value is 124o-02'-35" being the greatest angle giving a maximum difference of areas. For any other ratio of sides' the angle will be smaller.
For the ratio 3 -f- V 5 the angle is
10 120°. When the ratio is j ' the value of
o« + 6* . 101 Cos 2 C
3ab. is-jjQ andLog.-g-^-p is 0-52720, which corresponds to an angle
of about 4'-25 less than 105, or 104°-55-'75; and so in other cases. When the ratio of the sides becomes indefinitely great, the maximum difference angle approaches indefinitely near 90.
17. In well chosen triangles, there are not usually any very great differences in the sides, and hence practically the greatest differences will usually occur when C is not far from 120°.
18. If for example we suppose a triangle with sides of a degree each, and containing an angle of 120°, by the original formula the excess is27"-210 and the difference in area between the spherical and plane triangles is 0-18214 square miles, corresponding to an excess of O"-0024176. On a triangle with degree sides and the maximum angle of 124°02'-35" the excess is 26"-035 the differences of areas 0-18320 square miles, corresponding to an excess of 0"-0024318. Such differences though utterly innsible in the telescope, are still much greater than have ever occurred in practice; for though single sides of more than a degree be nothing very extraordinary, it is but rarely that two such sides can be found forming > triangle with a third side of from 118 to 120 miles.
19. The difference here treated of is, in similar triangles, proportional to the 4th powers of the homologous sides: Hence, in an equilateral with
half degree aides, this difference would be - of 0"-005245, or 0"-00006475;
and on the isosceles with half degree sides containing 120°, the difference
Roold be — of 0"-0024176, or 0"-00001511. Triangles such as these are not
very uncommon, but it is much more common to have triangles with less than half of their area.
20. It is thus fairly proved that the difference between the excess on a spherical triangle computed rigidly, and that deduced by reckoning its area u equal to that of a plane triangle of the same sides and contained angle, a a quantity so small that, even in extreme cases, the neglect of it will induce no sensible error; and that in triangles such as usually occur in practice, the difference is so utterly insignificant, that to go much out of the usual way in order to take account of it, would be a very needless refinement.
Sotes regarding the Meteorology and Climate of the Cape of Good Hope. By Robert Trotter, Esq.. Bengal Civil Service.
When last at the Cape it occurred to me, that a few particulars regarding the climate of a place, to which so many resort from this country in search of health, might be found interesting as well as useful: and particularly to medical men, by enabling them to judge how far it is likely to prove beneficial to those patients, for whom they may consider an absence from India necessary. If you deem the accompanying Meteorological Table, and the following cursory remarks worthy of a place in your Journal, I shall feel obliged by your inserting them.
The table contains an abstract I prepared from the Meteorological Registers of the Royal Observatory at the Cape, shewing the mean monthly weight and temperature of the atmosphere, and the minimum of each month for three years together, with the monthly fall of rain for the same period; and in order to compare the results with the climate of India, I have inserted corresponding observations made at Calcutta for an equal period, and likewise the monthly means of a year's observations at several other stations ; viz. Darjeeling, Dacca, and Cawnpore, extracted chiefly from the Journal of the Asiatic Society.
The Cape observations were made at 3 hrs. 15' P. M., being the period of least atmospherical pressure; the Thermometers hang on the Southeast side of the building, in the shade, and protected from solar radiation; 4 P. M. is the hour of most of the Indian observations, a few only of those at Darjeeling having been made at 4 hrs. 30' and 5 hrs. P. *.— the time of each set of observations therefore, being about an hour after the hottest period of the day, a rough estimate may be formed of the usual afternoon temperature, as well as a pretty fair comparison of the maximum temperature of the above places with that of the Cape, while from the means of the monthly minima, a comparison may be formed of the greatest average cold at the Cape and Darjeeling.
As Cape Town lies close to the base of Table Mountain, which, together with the Lion and the Devil's Peak encompasses it on three sides, its temperature is considerably higher than that of the Observatory which is nearly three miles distant, and being situated on the low isthmus between False Bay and Table Bay, enjoys the benefit of the breeze which generally blows from one bay or the other.
The Camp ground, Rondebosch, and Wynberg, possess a similar advantage in point of situation over Cape Town, (from which they are distant from 4 to 8 miles.) They are the favourite abode of Indian visitors during the warm months, but as they lie nearer than the Observatory to the mountain, the weather is much damper, and the fall of rain considerably greater during the winter, than at that place. In the hot weather, however, they certainly enjoy a cooler climate, in consequence probably of the greater abundance of verdure and shade.
Table Mountain, and indeed the whole range of hills, of which the Devil's Peak is the northern extremity, produce a variety of interesting atmospherical phenomena, and often times occasion an entire difference in the state of the weather at Cape Town, which is situated on the west side, and at Wynberg and Rondebosch on the other side of the range.
The north-west winds which prevail during the winter, are always loaded with much vapour, and bring much rain, but as the rain is frequently not formed till the vapour, after passing over Cape Town, has reached the cold summit of the mountain, it very often happens that though a fine day in Cape Town, it is raining heavily at Wynberg, Rondebosch, and other places on the lee side of the mountain. During the summer months, the same cause gives rise to a similar phenomenon, and occasions the well-known appearance on the top of the mountain, called the Table Cloth. The south-east sea breeze, which prevails at tlw