But if u represents the mean longitude of the sun, and P the parameter of the earth's orbit, we should have But also R2 du=√Pdt. sin sin i sin u, i being the inclination of the equator to the ecliptic: therefore the above expression becomes VP SSS. sin i sin u sin λ cos dudλdμ The limits between which the integrations for u and are to be effected, will depend on the figure of the surface under consideration. For simplicity, let it be an extremely small portion of the surface of the island included between two meridians, so close to each as to include a nearly rectangular space between their segments and those of the two isothermal lines. If m be the breadth of the rectangle, we may take u from o tom, and from, to,,, being the latitude of the northern extremity of whichever of the isothermals is nearest the coast, and 4, the latitude of the northern extremity of the other isothermal. The area under consideration will be m(2,-2). The sun's longitude u must be taken from 0 to 27 in estimating the amount of solar heat received during a year. 4. The heat received by the element m (2,-,) from the influences of causes, independent of direct solar radiation, will, as already stated, be a function of the distance of this element from the coast; it will therefore be a function of the difference of its latitude and that of the nearest point on the coast. If we make,+4=24, and represent the latitude of the northern part of the coast nearest the element of surface by 1, we shall have for H the proportion of heat received by the element during a year, the expression a2 Gm S H=ƒ(l—4) The second of these integrals vanishes between the limits, and the first may be determined by the properties of elliptic functions for E() representing a complete elliptic function of the second order, whose modulus is i. The value of i being 23° 28, E(i) =1.50658: consequently we may ultimately write -3 01316 a2 G H=ƒ(l−4) NP But as sin (2,-1)=2,-22, very approximately, this may be written, Similarly H, the proportion of heat received by the very small and nearly equal area included between the southern extremities of the isothermals, may be written where = and I, the latitude of the nearest point of the southern coast of the island. f(l−4) and ƒ (4,-1,) are both positive, and both are supposed, in virtue of what has been already stated, to possess the property of varying inversely with 1-4 and 4-1, respectively; in other words, f(-4) increases when 1-4 diminishes, and f(4,-1,) increases when 4,-1, diminishes. If we divide H and H, by the nearly equal areas m (2,-2) and m (2,-2) respectively, the results will represent the amount of heat received by the units of surface at the northern and southern extremities of the isothermals. These quantities should be equal; hence we shall have, very approximately, 3 cos 2 4+ƒ (l−1) = cos 2 4, +ƒ (41−l1); But as cos 24<cos 24,, it follows that f(1-4)>f(4,-1,), and, consequently, 1-4<4,-1,. If the influence of solar radiation were not considered, these quantities would be equal: consequently its tendency is to transport the closed isothermal line from south to north, by making the distance of its northern extremity from the northern coast less than the distance of its southern extremity from the southern coast. The same result will affect the next adjacent isothermal, and so on in succession, so that ultimately all the isothermal lines will be transported towards the north. * Poisson gives 0·17798, as its logarithm from Legendre, Théorie de la Chaleur, p. 490. the heat received at any point of the earth's surface from solar radiation alone, abstracting the influence of atmospheric absorption in different latitudes, varies in conformity with Mayer's law as the square of the cosine of the latitude. The more the influence of latitude predominates over all other causes, the more will the positions of the isothermals be changed. in the manner above indicated: it follows, therefore, that while towards the equatorial coast of an island these lines terminate on the coast, they may still continue as closed curves in the interior of the island. If the influence of differences of latitude was greatly predominant over all other climatic influences, all the isothermals may terminate on the coast. 5. The quantity of heat received by a given small area during the summer and winter half-years, between the spring and autumnal equinoxes, may be readily found by integrating with respect to u, within the limits 27 and 7, and afterwards within the limits and 0. Thus we shall have the general expression the term affected by 2 sin i is to be taken with the positive sign for that half of the year during which the sun is at the same side of the equator as the area in question, and the negative sign for the other half of the year. If ,-, be so small that its square may be neglected, then for the small area s = m (^, -22) we shall have the amount of solar heat H, received during either half year expressed by the equation H1= K(E(i) cos2±sin i sin 24), making K= 2 a2 Gs (5.) sin 24 is always positive, as 4 cannot exceed 90°, it follows, therefore, that the influence of latitude on the points of the isothermals will be greater during the summer half of the year than during the winter half; and therefore, all other things remaining the same, the isochimenal lines, or lines of equal winter temperature, would be less displaced from their concentric position in an island than the isotheral lines, or lines of equal summer temperature. From the preceding expression we can determine the latitude of the parallel which receives the greatest amount of solar heat during the summer half of the year. For on differentiating we have If in (6) we substitute the values of E() and sin i respectively, we shall find 4= 7° 24′ nearly, cos 24 and sin 24 will both be d2 H d 12 positive, and therefore negative; the above value of tang. 24, gives therefore a maximum value to H, and consequently the parallel which receives the greatest amount of solar heat during the half year that the sun is at the same side of the equator, is the parallel which has the latitude 7° 24'. 6. The results of these investigations become applicable to the two great continents of the eastern and western hemispheres; for as these are both completely surrounded by water, they may be considered as two immense islands. The distance from the ocean of the greater part of their surfaces, diminishes so much the action on their general climate of the waters by which they are surrounded, that the influence of difference of latitude becomes as a general rule, predominant over all other causes, and the centres of most of their isothermal lines are transported so far towards the pole, that many of these lines circumscribe the earth's axis, or lie in surfaces which cut that axis more or less obliquely. In the interior of a continent, an elevated table-land of limited dimensions is circumstanced nearly in the same way as an island, for its edges are surrounded with air having a mean temperature nearly uniform, and different from that lying on its surface. We may therefore expect to find, even in the interior of continents, closed isothermal lines, as well as in the interior of oceanic islands. The disturbing action of general winds will modify the forms of the isothermal lines, according to the frequency and the temperature of these winds. The warm winds will cause the isothermals to recede from the coast towards the interior in a direction opposed to that from which they emanate; the cold winds will, on the contrary, cause the isothermals to advance towards the direction from which they blow. We may, therefore, conceive the tendency of such general winds, when warm, to be to remove the centres of the isothermals from the points whence they blow; when cold, their tendency will be to approach these centres towards the same points. If we compound these tendencies with the effect of differences of latitude, we would have the resultant direction towards which the isothermal lines should be displaced from their concentric position by the action of all these disturbing causes. ART. XXXVI.-On the possible Intersection of the orbits of Mars and certain of the Asteroids; by Professor DANIEL KIRKWOOD, of the Indiana University. THE present eccentricities of the asteroidal orbits are included between the limits 0.046085 and 0.336987. Of these, some are increasing, others diminishing. We are not aware, however, that the range of variation has, in any instance, been accurately determined. If we assume the superior limit of the eccentricity, in the case of the following members of the group, to be 0.25, (and this is less than the present eccentricity of Juno, Phocæa, Polyhymnia, and Atalanta,) their perihelion distances at the epochs of maximum eccentricity will be as follows: Flora, 1.650940 1.652879 1.700361 1.722045. The present aphelion distance of Mars is 1.665725, the eccentricity of the Martial orbit is, however, increasing; the secular variation being 0.000090176. According to LeVerrier the maximum eccentricity will be 0.14224. The corresponding aphelion distance will be 1-740431; greater than the least perihelion distances of the asteroids above named. It is obvious therefore that if the longitudes of Mars and any one of these bodies should differ by nearly 180° when the eccentricities of both are not far from their superior limits, the orbits or at least their projections on the plane of the ecliptic, must intersect. When it is remembered that the variation of the eccentricity is extremely slow, that the line of apsides of the orbit of Mars completes a revolution in less than 20,000 years, aud that the inclinations of the orbits of Flora, Harmonia and Ariadne, are small, the probability of a very near approach of Mars and some of these small planets an approach so close as to render the question of the perturbations of the latter both curious and interesting is at once apparent. If we assume the greatest eccentricity now found in the group, as the superior limit of the variation of all, the maximum aphelion distance of Mars will be greater than the minimum perihelion distances of twenty of the small planets. Bloomington, Ind., Feb. 11, 1859. |