521 



GRAVITATION. 



GREAT CIRCLE OR TANGENT SAILING. 



regressing the quicker. In point of fact, the circumstances of the 

 inner satellites are very nearly the same as if no other disturbing force 

 existed, so great is the effect produced by Jupiter's ob lateness. 



(240.) The figure of Saturn, including in our consideration the ring 

 which surrounds him, is different from that of Jupiter ; but the same 

 principles will apply to the geueral explanation of its effects on the 

 motion of its satellites. The body of Saturn is oblate, and the forces 

 which it produces are exactly similar to those produced by Jupiter. 

 The effect of the ring may be thus conceived : If we inscribe a 

 spherical surface in an oblate spheroid, touching its surface at the two 

 poles, the spheroid will be divided into two parts ; a sphere whose 

 attraction is the same as if all its matter were collected at its centre, 

 and an equatorial protuberance analogous in form to a ring. The whole 

 irregularity in the attraction of the spheroid is evidently due to the 

 attraction of this ring-like protuberance, since there is no such irregu- 

 larity in the attraction of the sphere. We infer therefore that the 

 irregularity in the attraction of a ring is of the same kind as the 

 irregularity in the attraction of a spheroid, but that it bears a much 

 greater proportion to the whole attraction for the ring than for the 

 spheroid, since the ring produces all the irregularity without the whole 

 attraction. Now, the plane of Saturn's ring coincides with the plane 

 of Saturn's equator, so that the effect of the body aud ring together is 

 found by simply adding effects of the same kind, and is the same as if 

 Saturn were very oblate. The rate of progression of the perisaturnium 

 of any satellite, and the rate of regression of its node, will therefore be 

 rapid. In other respects it is probable that the theory of these satellites 

 would be very simple, since all (except .the sixth) appear to be very 

 small, and the sun's disturbing force is too small to produce any 

 sensible effects. 



(241.) The satellites of Saturn, except the sixth, have been observed 

 so little that no materials exist upon which a theory can be founded. 

 A careful series of observations on the sixth satellite has lately been 

 made by Bessel, from which, by comparing the observed progress of 

 the perisaturnium and regression of the node with those calculated on 

 an assumed mass of the ring, the real mass of the ring has been found. 

 It appears, thus, that the mass of the ring (supposing the whole effect 

 due to the ring) is about y'.th of the mass of the planet. 



(242.) The effect of the earth's oblatene&s in increasing the rapidity 

 of regression of the moon's nodes is so small, that it cannot be dis- 

 covered from observation. But the effect on the position of the funda- 

 mental plane is discoverable. We have seen (204) that the moon's 

 line of nodes regresses completely round in 19 J years. The plane of 

 the earth's equator U inclined 23 J to the earth's orbit, and the line of 

 intersection alters very slowly. At some time therefore the line of 

 nodes coincides with the intersection of the plane of the earth's 

 equator and the plane of the earth's orbit, so that the plane of the 

 moon's orbit lies between those two planes ; and 9] years later, the 

 line of nodes again coincides with the same line, but the orbit is 

 inclined the other way, so that the plane of the moon's orbit is more 

 inclined than the plane of the earth's orbit to the plane of the earth's 

 equator. Now it is found that in the former case the inclination of 

 the moon's orbit to the earth's orbit U greater than in the latter by 

 about 16", and this shows that the plane to which the inclination has 

 been uniform is neither the plane of the earth's equator nor that of the 

 earth's orbit, but makes with the latter an angle of about 8", and is 

 inclined towards the former. 



(243.) There U another effect of the earth's oblateness (the only 

 other effect on the moon which is sensible) that deserves notice. The 

 inclination of the moon's orbit to the earth's orbit is less than 5, and 

 the inclination of the earth's equator to the earth's orbit is 234. 

 Consequently, when the moon's orbit lies between these two planes, 

 the inclination of the moon's orbit to the earth's equator is about 19 ; 

 and when the line of nodes is again in the same position, but the orbit 

 is inclined the other way, the inclination of the moon's orbit to the 

 earth's equator is about) 28. At the latter time, therefore, in conse- 

 quence of the earth's oblateness, the moon, when farthest from its 

 node, will, by (235), experience a smaller attraction to the earth than 

 at the former time when farthest from its node. When in the line of 

 nodes, the attractions in the two cases will be equal. On the whole 

 therefore the attraction to the earth will be less at the latter time than 

 at the former. For the period of 9f years therefore the earth's attrac- 

 tion on the moon is gradually diminished, and then is gradually 

 increased for the same time. The moon's orbit (47) becomes gradually 

 larger in the first of these times, and smaller in the second. The 

 change ia very minute, but, as explained in (49) the alteration in the 

 longitude may be sensible. It is found by observation to amount to 

 about 8", by which the moon is sometimes before her mean place, and 

 sometimes behind it. If the earth's flattening at each pole were more 

 or'less than jijth of the semi-diameter, the effects on the moon, 

 both in altering the position of the fundamental plane, and in pro- 

 'lu'-ing this inequality in the longitude, would be greater or less than 

 the quantities that we have mentioned ; and thus we are led to the 

 very remarkable conclusion, that by observing the moon we can discover 

 the amount of the earth's oblateness, supposing the theory to be true. 

 Thin has been done ; and the agreement of the result thus obtained, 

 with that obtained from direct measures of the earth, is one of the 

 most striking proofs of the correctness of the Theory of Universal 

 Gravitation. 



GRAVITY, CENTRE OF, is that point at which all the weight of a 

 mass might be collected without disturbing the equilibrium of any 

 system of which the mass forms a part. Thus, if a lever were balanced 

 jy means of two solid spheres of uniform density hung at the ends, 

 ;he equilibrium would still remain if all the matter of either of the 

 spheres could be concentrated at its centre. The centre of the sphere 

 is then its centre of gravity. 



When a body is suspended by a string, and allowed to find its 

 position of rest, the centre of gravity is in the line of continuation of 

 the string. If then a body be suspended successively at two different 

 points, and if the lines in which the strings produced would cut 

 through the body can be conveniently determined, the centre of 

 gravity is the point of intersection of the two lines. This process is 

 very easy in the case of a thin flat surface, and the approximation is 

 quite sufficient for practical purposes. 



When a surface (or a thin plate) is of uniform density, the centres 

 of gravity and of figure [CENTRE] are the same. It is needless to say 

 where this falls in the case of a circle, of a square or other parallelogram, 

 or of a regular oval figure. In a triangle it is found by joining the 

 vertex and middle of the base, and cutting the intercepted line into 

 three equal parts, the nearest trisecting point to the base giving the 

 centre of gravity. In a prism and cylinder it is the middle point of 

 the line joining the centres of gravity of the two bases. In a cone or 

 pyramid it is found by joining the vertex and the centre of gravity of 

 the base, and cutting the joining line into four equal parts, the nearest 

 of which to the base ends in the centre of gravity. In a semicircle the 

 distance of the centre of gravity from the centre is about fourteen 

 thirty-thirds of the radius ; in a hemisphere this same distance is five- 

 eighths of the radius. 



The centre of gravity of two bodies is found by joining their centres 

 of gravity, and dividing the joining line so that the content of the first 

 may be to that of the second as the segment adjoining the second is to 

 that adjoining the first. By the same rule, and by the centre of two 

 bodies thus found and that of a third, the centre of three bodies may 

 be found, taking care to use with each centre the sum of all the 

 contents of the bodies employed in finding it. 

 GRAVITY, SPECIFIC. [SPECIFIC GRAVITY.] 

 GREAT CIRCLE or TANGENT SAILING. Although the subject 

 will be found properly introduced under the general head Navigation, 

 the peculiar nature of Great Circle Sailing, and its growing importance 

 to underwriters, shipowners, and therefore to the whole mercantile 

 interests of Great Britain, demand a special notice in this place. 



The opening of an expanding and lucrative traffic with the far east 

 of the globe, has called more imperatively for improvements in science 

 and art, which the merchants of old, in their occasionally distant 

 voyages, the less needed. To them indeed one item of daily expense 

 was unknown ; the consumption of coal had not advanced the costs 

 of a voyage to an enormous outlay, nor had the wear and tear of 

 machinery entailed on shipowners disbursements which at this day 

 form so prominent a feature in their aggregate yearly expenditure. 



It was to meet this and to render facilities for shortening a voyage 

 already diminished in duration by the use of steam, that attention was 

 attracted to the long neglected practice of great circle sailing. 



To explain the principles of this sailing it is only necessary to take 

 an ordinary terrestrial globe, and notice that while some circles usually 

 drawn thereon and called meridians are of equal size, because if the 

 globe were cut in two in the plane of any one of these meridians, the 

 section would pass through the globe's centre, other circles called 

 parallels (as of latitude, declination, altitude, &c.) differ in magnitude. 

 The term great circle only applies to the former : a sphere therefore 

 being cut through on a great circle would be divided into two equal 

 parts. The equator is of this description. If we take a thread and 

 stretch along the equator, say reaching along 80 or 90 degrees of 

 longitude, on tightening the thread it would still lie exactly along the 

 line : such would also be the case on the meridians ; but if this be 

 attempted on any of the parallel circles, the thread on being tightened 

 would slip off the parallel in a direction towards the nearest pole, thus 

 forming an illustration of a great circle track ; and the farther from 

 the equator this is tried, the more striking would be the difference 

 between the parallel and the track, which thus measures the nearest 

 distance between the ends of the thread, and this will be the case 

 wherever a thread is thus applied obliquely on a sphere. 



A general belief prevails among merely practical men that this kind 

 of sailing is one of recent development that its employment has 

 been in consequence of modern discovery ; but such is far from the 

 truth. Those who have studied the practice of navigation as given in 

 old authors will be aware that for nearly 1 50 years before the dawn of 

 this century a knowledge of great circle sailing was not only considered 

 to be a necessary element in the acquirements of a navigator, but a 

 knowledge of its very principles precisely as laid down in the most 

 modern works (so far as regards construction and calculation) was 

 thought an essential part of the educational course of every sea-going 

 merchant captain, and its rules and demonstrations are to be found 

 plainly Bet forth in such works as Atkinson's, Norwood's, Newhouse's, 

 Gallibrand's, and others of the period of about 1680 to 1720. 



Spherical trigonometry was naturally considered indispensable to 

 every one employed in conducting treasure over the surface of a 

 2>hcrc, while a knowledge of plane trigonometry was deemed an equally 



