marble mosaic, from the Roman and Alexandrian work It will be seen, indeed, from these few instances, how greatly the elements of length and height increased in after times. That of cubic capacity, on the contrary, seems to have diminished. If we deduct from the medieval churches the parts walled off by solid stone screens or wooden partitions, and the immense space taken up by the piers necessary for the support of vaulted roofing, we shall see how decidedly the older buildings have the advantage, as far as the question goes of holding vast multitudes at a time. A simpler arrangement than that of the western basilica can hardly be conceived, and, graceful and wonderful as the subsequent architecture became in southern as well as northern Europe, no buildings whatever have surpassed these early churches either in solemnity or practical convenience. JOHN H. POLLEN. 145 SCIENTIFIC RESEARCHES. ART. I.-On the Influence of the Great Inequalities of IN duct of lities of N calculating the amount of the perturbation in the The propath of one planet caused by the action of another, the it is usual to assume that both the planets would, except turbing for the inequalities caused by their mutual attraction, two move accurately in ellipses, no account being taken of planets the small inequalities in the path of either of them caused causes by a third planet. But since in reality any alteration in inequa the place of either of them causes an alteration in their mutual distance, it is evident, according to the theory riod. of gravity, that their mutual attraction will be altered also; that is, that their disturbing force will be different; or in other words, that a new disturbing force will be added, which will produce its corresponding inequality in the path of each of them. Now, the undisturbed place of a planet so nearly coincides with its actual place, that we may in most cases altogether neglect any difference in the disturbing force which arises from the planet not being exactly in the place where it would be if undisturbed; but we cannot do so always. For example, in the case of Saturn, there exists an inequality in longitude which is represented by 652" sin (2n,t-4n,t+II), together with another of about half the magnitude of the radius vector. These, in linear measure at least, are about equal to the eccentricity of the Earth's orbit; and therefore, under certain circumstances, are quite large enough to produce a sensible inequality in the motion of a neighbouring planet. In fact, when we remember that the effect of a disturbing force depends not only upon its intensity, but also, in some cases, upon the square of the time during which it acts in one direction, it is evident that a very minute force, if its period is very long, may produce very sensible results. In order therefore to complete the theory of planetary long pe perturbations, it becomes quite necessary to search out all forces of this kind whose period is very long, or in other words, to examine whether in the formation of the differential equations of motion, we may not by the employment of the true coördinates of the planets, find terms in the disturbing function which we should not find by merely using the elliptic ones, having a very small divisor, and which consequently will be much increased by integration. For example, suppose that we are calculating the mutual perturbations of Saturn and Uranus. Now, if we employ the terms above mentioned, which occur in the true coördinates of Saturn, it is evident that we shall obtain a term in the differential equations of motion of the form Bsin (3nt-6nt+2n ̧t+ПI) n, no, no, being the mean motions of Jupiter, Saturn, and Uranus. This expression, as it will have to be twice integrated, will be twice divided by 3n-6n+2n,, which is a small quantity; and the term which results in the longitude of Uranus from the employment of the above mentioned and other inequalities which will produce terms of the same form, amounts to about 43 seconds. This, though much smaller than some of the inequalities in the path of Uranus, is still considerable with respect to many that occur in the planetary theory, being in fact larger than any that have been hitherto noticed in the motions of Mercury, Venus, the Earth, or Mars; and being more than five times as large as any of the perturbations of the Earth. Its period is about 1,700 years; and there will be corresponding terms in the longitude both of Saturn and Jupiter; that of Saturn amounts to about 40", and that of Jupiter to about 10′′. InequaAnother remarkable inequality of the kind which I lity of have found, is one which appears to exist in the motions long pe- of the Earth and Mars.' Its period is somewhat longer riod in 1 Since the above has been in type, we have found that this inequality, as far at least as the motion of the Earth is concerned, has been already calculated by MM. Leverrier and Hansen. The coefficient given by the former (see vol. xxx., p. 463, Comptes rendus) is 7.162”. That by the latter is 6.221". That here obtained is 7.293", which therefore agrees very closely with that given by M. Leverrier. Also the small inequality noticed further on, which depends on the disturbing forces of Mars and Venus, has been given by the latter writer; but not that depending upon the product of the forces of Jupiter and Saturn. Mars than the one mentioned above, being about 1,800 years, the moor about twice that of the long inequality of Jupiter and tons of Saturn. It arises in the following way: Four times the and the mean motion of the Earth is very nearly equal to eight Earth. times that of Mars minus three times that of Jupiter; it has an exceedingly small divisor; and in fact the term which represents it in the disturbing function is multiplied by a quantity which amounts to upwards of twenty millions! This inequality appears to amount, in the case of the Earth, to about 7.29", and in that of Mars to about 45"; quantities which are larger, and in the case of Mars very considerably so, than any which arise from simple perturbation; the largest hitherto known in the motion of the Earth, amounting to not more than 7.15", and in the motion of Mars to 24.53". One remarkable thing about this inequality is the effect which it appears likely to have upon the motion of the Moon. According to the investigations of Professor Hansen, the inequality lately discovered by Mr. Airy in the motion of the Earth, which amounts to about 2.05′′ with a period of 240 years, gives rise to an inequality in the motion of the Moon amounting to about 23′′; and therefore it would seem that there will also be a corresponding one arising from that above mentioned, and which will amount to something considerable. I have not been able to get access to Professor Hansen's investigations, and have not yet finished the calculation of the lunar inequality, and therefore cannot as yet give its precise amount; but I hope to have done so in time to publish the results in the second number of this Periodical. And, in general, in order to find quantities of this kind, we must not only seek out all terms in which Pin-pin, is small, n1, ng, being the mean motions of the planets, and P1, P2, ..., any whole numbers, but also all those in which pn-p,n,p,n, is small with respect to any of the three quantities n1, n,, ng, for all such terms will be much increased by integration. It will be seen that all such quantities will be of the order of the second power, or rather of the product of two powers of the disturbing force. It should also be observed, that in general, if p, n, relate to that one of the three planets which lies between the other two, the quantity p, ought to have a different sign from p, and p,, in the same way that where two planets only are concerned, p, and p, should have different signs; otherwise such high powers of the eccen |