GRAVITATION. 



GRAVITATION. 



/p. 2*. I1 th condition, will be attuned. For the Telocity at a, is 

 diminished by the disturbing fore* having acted perpendicularly to 



Fir. 16. 



the radius rector, while the moon BOM from n, to n, ; and though, 

 the distance from A being greater, the earth's attraction at a, will bo 

 less than the attraction at a, ; yet, when increased l>y the disturliiiiK 

 force, directed to A at B,, it will be very little less than the att 

 diminished by the disturbing force at a,. The diminution of velocity 

 then at a, being considerable, and the diminution of force small, the 

 curvature will be increased ; and this increase of curvature, by proper 

 choice of the proportions of the oval, may be precisely such as cor- 

 responds to the real difference of curvature in the different parts of 

 the oval. Hence, such an oral may be described by the moon with- 

 out alteration in successive revolution*. 



(&2.) We have here supposed the earth to be stationary with respect 

 to the sun. II however we take the true case of the earth moving 

 round the sun, or the sun appearing to move round the earth, we have 

 only to suppose that the oval twists round after the sun, and the same 

 reasoning applies. The curve described by the moon is then such AS 

 is represented infg. 27. As the disturbing force, perpendicular to tin- 



i 



radius vector, acts in the same direction for a longer time than in the 

 former case, the difference in the velocity at syzygies and at quadra- 

 tures is greater than in the former case, and this will require the oval 

 to differ from a circle, rather more thtui if the sun be supposed to 

 stand still. 



^93.) If, now, in such an orbit as we have mentioned, the law of 

 ni/urm drtcriplion of artat ly the radiu* rector were followed, as it 

 would be if there were no force perpendicular to the radius vector, the 

 angular motion of the moon near a, and B,, Jig. 26, would be much 

 less than that near a, and a,. But in consequence of the disturbing 

 force, perpendicular to the radius vector (which retards the moon from 

 B, to B,, and from B, to a,, and accelerates it from B a to a,, and from 

 a, to a,), the angular motion is still less at B, and a., and still greater 

 at a, and a,. The angular motion therefore diminishes considerably 

 while the moon moves from B, to a,, aud increases considerably while 

 it moves from B, to B,, Ac. The mean angular motion, determined 

 by observation, is less than the former and greater than the latter. 

 Consequently, the angular motion at B, is greater than the mean, and 

 that at a, is less than the mean ; and therefore (as in (90),) from a, to 

 a. the moon's true place is before the mean ; from B, to B, the true 

 place is behind the mean ; from B, to a, the true place is before the 

 mean ; and from B, to a, the true place is behind the mean. This 

 inequality is called the moon's variation ; it amounts to about 32', by 

 which the moon's true place is sometimes before and sometimes behind 

 the mean place. It was discovered by Tycho Brahe, from observation 

 about 1590. 



(U4.) We have however mentioned, in (79), that the disturbing 

 forces are not exactly equal on the side of the orbit which is next the 

 sun, and on that which is farthest from the sun ; the former being 

 rather greater. To take account of the effects of this difference, let 

 us suppose, that in the investigation just finished, we use a mean value 

 of the disturbing force. Then we must, to represent the real case, 

 suppose the disturbing force near conjunction to be increased, and 

 that near opposition to bo diminished. Observing what the nature of 

 these forces is (77), (78), and (84), this amounts to supposing that near 

 conjunction the force necessary to make up the difference is a force 

 acting in the radius vector, ana directed from the earth, and a force 

 perpendicular to the radius vector, accelerating the moon before con- 

 junction, and retarding her after it, aud that near opposition the forces 

 are exactly of the contrary kind. Let us then lay aside the considera- 

 tion of all other disturbing forces, and consider the inequality \\lii.-li 

 these forces alone will produce. As they are very small, they will nut 

 in one revolution alter the orbit sensibly from an elliptic form. What 

 tlii-n must be the excentricity, and what the position of the line of 

 apses that, with these disturbing forces only, the same kin. I of oil.it 

 may always be described f A very little consideration of (57), (58), 

 and (08), will show, that unless tile line of apses) oast through the sun, 

 the vxocntrii ity will either be increasing or diminishing from tin- 

 action of these forces. We must assume therefore, as our orbit is to 

 have the same excentricity at each revolution, that the line of apses 

 passes throng), tlic sun. But is the perigee or the apogee to be turned 

 towards the sun ? To answer thin question we have only to observe 

 that the lines of apses must progress as fast a- i-oars to pro' 



gress, and we must therefore choose that position in which the forces 

 will cause progression of the line of apses. If the perigee be directed 

 to the sun, then the forces at both parts of the orbit will, by (51), (54), 

 (65), and (66), cause the line of apses to regress. This supposition, 

 then, cannot be admitted. But if the apogee be directed to the sun, 

 the forces at both parts of the orbit will cause it to progress ; and by 

 (56), if a proper value is given to the excentricity it will progress 

 exactly as fast as the sun appears to progress. The effect, then, of the 

 difference of forces of which we hare spoken, is to elongate the orbit 

 towards the sun, aud to compress it on the opposite side. This irregu- 

 larity is called the parallaelic inequality. 



We shall shortly show, that if the moon revolved in such an elliptic 

 orbit as we have mentioned, the effect of the other disturbing forces 

 (independent of that discussed here) would be to make its line of apse* 

 progress with a considerable velocity. The force considered here, 

 therefore, will merely have to cause a progression which, added to that 

 juat mentioned, will equal the sun's apparent motion round the earth. 

 The excentricity of the ellipse, in which it could produce thin smaller 

 motion, will (56) be greater than that of the ellipse in which the same 

 force could produce the whole motion. Thus the magnitude of the 

 parallactic inequality is considerably increased by the indirect effect of 

 Vr disturbing forces. 



(95.) The magnitude of the forces concerned here is about jJoth of 

 those concerned in (91), Ac. ; but the" effect is about ^th of their effect. 

 This is a striking instance of the difference of proportions, in forces 

 anil the effects that they produce, depending on the difference in their 

 modes of action. The inequality here discussed is a very 

 one, from the circumstance that it enables us to determine with con- 

 siderable accuracy the proportion of the sun's distance to the moon's 

 distance, which none of the others will do, as it is found upon calcula- 

 tion that their magnitude depends upon nothing but the exoentrioities 

 and the proportion of the periodic times, all which are known without 

 knowing the proportion of distances. 



(96.) The effect of this, it will be readily understood, is to be com- 

 bined with that already found. [See the Note to (134).] The moon's 

 orbit therefore is more flattened on the aide farthest from the sun, 

 and less flattened on the side next the sun, than we foun-l in ('.' 

 (92). The equable description of areas is scarcely affected by these 

 forces. The moon's variation therefore is somewhat diminished near 

 conjunction, and is somewhat increased near opposition. 



(l>7.) It will easily be imagined, that if there is an excentricity in 

 the moon's orbit, the effect of the variation upon that orbit will lie 

 almost exactly the same as if there were no excentricity.* 

 supposing that the orbit without the disturbing force had such a form 

 as the dark line in .fig. 28, it will, with the disturbing force, have such 



Fig. 28. 



a form as the dotted line in that figure. The same must be under- 

 stood in many other cases of different inequalities which affect the 

 motion of the same body. 



As this general proposition li of considerable importance, we shall point 

 out the nature of the reasoning by which (with proper alteration For different 

 esses), the reader may ftti-fy himself of Its correctness. The reason wliy. In 

 fig. 29, the moon cannot describe the circle B,, 4,, B,, 6,, though it touches 



B! and B a , and the reason that it will describe the oval B,, B,, B 3 , B , In, 



1'ig. 20. 



that the disturbing force makes the forces at B, and B, less than they would 

 otherwise have been, and greater at B, nd B. than they would otherwise hare 

 been; and the velocity Is, by that part of the force perpendicular to the nullus 



made lens at B, than It would otherwise hare been. So that, unless we 

 i.npposed it moving at B, with a greater Telocity than It would have had nndti- 

 turbrd In thr cln-lr B,, ,, 4 , the gre.t curvature produced by the great 

 force and diminished Telocity at B, would have brought It much near, 

 than the point B, ; but wlih this large Telocity at B,, It will go out further at 

 B,, and then the great curTalure may mak it pass exactly through B,. In 

 like manner, In ft. SO, if the velocity at D, were not greater than It would 

 have had undisturbed In the elllpe B,, a,, t., the Increased curvature at B,, 

 produced by the increased force and diminished velot-ity time, Mould have 

 brought It n.ntb nearer to A than the point B, ; but wilh a lame velocity at *, 

 It will go out at B, further than It would otherwlw have (mnr out, and then 



. ed force and diminished velocity will cut TO Its course w> mu.h, that 

 It may touch the elliptic orbit at B, ; and on. The whole explanation, la 



