Ill 



GRAVITATION. 



GRAVITATION. 



Bl 



(188.) But though the alteration which the inclination produce* in 

 the force* that tend to disturb the body'* motion I'M it* plane may, in 

 most *e*, be neglected, yet the force which tend* to pull the body 

 6otr the plane, or Mow the plane, cannot be neglected. In almost 

 every cue thi* force will be lex than the force tending to disturb the 

 motion in the plane, yet it will be much greater than the alteration 

 which the inclination product* in that force. It is our object in this 

 Motion to show the nature of the alteration which is produced by the 

 force tending to pull the body from the plane. 



(189.) Find, thru, an to the effect of a force generally which act* 

 perpendicularly to the plane of revolution. (We shall confine our- 

 selves at present to force* which act perpendicularly to the plane, 

 became it U evident that forces which act in, or parallel to, the plane 

 of the orbit, whether in the radius vector or perpendicularly to it, will 

 not cause the planet to depart from that plane.) Let Jiy. 40 be a 

 perspective representation of an orbit, and a plane of n 

 Suppose M A N to be the line of node* at which the plane of the orbit 

 H , B, crone* the plane of reference D E ; the central body A being in 



Fie. 40. 



the line of nodes, and the part of the orbit marked by a dark line 

 being above the plane, and that marked by a dotted line being below 

 it. Suppose that the planet has moved from x to B,, and that at B,, 

 before it reached the point highest above the plane D K, a force pulls it 

 down towards the plane. After n short time, instead of going to B !F 

 where it would have been if no force had disturbed it, it will be found 

 at 4,, having described B, 6., instead of B, B.. It is plain that the orbit 

 in which the planet must have moved without a disturbing force, in 

 order to describe B,4, now, could not be SB,, but must be such a 

 curve as n B, , crossing the plane n K at a point in the situation of the 

 point i. Therefore, if no more disturbing force acts, the planet, 

 which lias described 8,4,08 if it came without disturbance from M, 

 will go on to describe an orbit as if it had come without disturbance 

 from n, and will therefore describe an orbit B, &.,, crossing the 

 plane E p in the point* n and m. The line of nodes is changed from 

 MAN to mA*. 



(190.) Here the line of node* has twisted in a direction opposite to 

 the planet's motion, or has regretted. The inclination of the new 

 plane U evidently legs than that of the old one, since it passes through 

 the same point B, and cuts the plane of reference in a line more distant 

 from B than the line in which the old one cut it, or the inclination is 

 iliminiilietl. 



(191.) Now, if we conceive that at B, (fig. 41), after the planet has 

 parnind the point highest above the plane, a force tends to pull it 

 towards the plane, the planet, instead of going to B 4 , will go to 4 4 , and 



Fig. 41. 



he action of the force while the planet i* in that part of it* orbit 

 which i* on the other side of the plane D B. 



We shall now proceed with the consideration of the force perpen- 

 jcular to the orbit, which i* produced by the attraction of a disturbing 



(195.) First : it is plain that, if the disturbing body is in the plane 

 if the orbit (produced, if necessary), it will not tend to draw i-itluT 

 he central body or the planet out of tint plaiir, iuul then-fore will 

 >roduce no disturbing force perpendicular to il.- p'.m ..i ili.-oil.it. 

 'roceeding, then, with the supi>oBitiou that the disturbing body is n.-t 

 in the plane of the orbit ; and supposing jig. 42 to be a perspective 

 view of on orbit B, B, B., (which, to assist our ideas, may be conceived 

 to differ little from a circle) with the disturbing body c out of the 



Fig. 42. 



instead of crossing the plane D E at M, will cross it at m ; and then, il 

 it is not disturbed again, will proceed in an orbit of wliic.h n, h t m is a 

 part, and which will cross the plane D E at the points m and . The 

 new line of node* has twisted here also in the direction opposite to 

 the direction of the planet's motion, or ho* regretted. But the 

 inclination of the new plane is greater than that of the old one, since 

 it puses through the same point B,, and cut* the plane of reference in 

 a line lea* distant from B, than the line in which the old one cut it, or 

 the inclination i* incnated. 



(192.) We have, then, this general result : If a force acting perpen 

 dicularly to the orbit tend* to draw the planet toward* the plane o: 



while "the*plianer'i]r moving from the highest point to% node, it 

 increase* the inclination. 



(198.) In the same manner, if the force tends to draw the planet 

 from the plane of reference, it always causes the line of node* to 

 progress. While the planet is moving from a node to the pom 

 highest above the plane, it increases the inclination ; anil w -hilr th 

 planet U moving from the highest point to the node, it diminishes th 

 inclination. 



(194.) Similar result* would have been obtained if we had considered 



ilanc of the orbit, let us take three points B, B, B,, of which n, is at 

 ihe same distance as A from c, B, U nearer to c, and B, farther from o 

 .li.ni A is. Suppose that the attraction of o draws A in a certain small 

 ,ime through the space A a, and that when the planet is at B,, or B., 

 or B,, the attraction draws the planet in the same time through B,4 1( 

 or B, 4,, or B, 4, respectively. Then (as in (71)) the attraction of c 

 upon the two bodies A and B would produce no disturbance in their 

 relative motions, if it drew them through equal spaces in the some 

 direction. Draw B,rf,, B,(i,, B s rf s each equal and parallel to A a ; then 

 if the attraction had drawn B, to rf,, there would have been n 

 turbance, and consequently the real disturbance at B, U represented 

 by a force which would have drawn the planet from rf, to 4.. Similarly, 

 the real disturbances at B, and B, are represented by forces which 

 would have drawn the planet from tl, to 6,, and from d s to 4 a 

 respectively. Now, since CB, is equal to CA, the forces of c upon A 

 and B, are equal, and therefore B,4, is equal to A a, and tin 

 a 4, is parallel to A B,, and therefore is in the same straight line 

 with 6, f/, ; and consequently at B, the whole disturbing fm 

 parallel to the radius vector, and there is no port perpendicular to the 

 plane of the orbit. But at B a the planet is nearer to c, the force 

 therefore on the planet is greater, and B, 4, is therefore greater than 

 A a or B, d t ; also it is more nearly perpendicular to the plane of the 

 orbit than B,</_ ; and consequently 4, is farther from the plane of 

 the orbit than a t ; and therefore the disturbing force rf, 4 ? is directed 

 from the plane of the orbit towards the side on which c is. On the 

 contrary, at B, the planet is farther from c ; the force on the pi 

 therefore less : and B, 4, is therefore less than A a or B, rf, ; moreover 



we find, 



(196.) When the central and revolving bodies are equally distant 

 from the disturbing body, there is no disturbing force perpendicular to 

 the plane of the orbit. 



(197.) When the revolving body is nearer the dMarUng body than 

 the central body is, the disturbing force perpendicular to the plane 

 tends to draw the revolving body out of the plane to tlmt xiilc on 

 which the disturbing body is. 



(198.) When the revolving body is farther from the disturbing Ixx3y 

 than the central body is, the disturbing force perpendicular to the 

 plane tends to draw the revolving body out of the plane to tin 

 opposite the disturbing body. 



We may now apply these conclusions to the alteration of the midl- 

 and inclination i.f tin moon's orbit produced l>y the MIII'S tt> 

 The plane of reference is here supposed to be the plane of tin- 

 orbit. 



(199.) First : suppose the line of nodes of the moon's orbit to be 

 in syzygie*, or to pass through the sun. Here the sun is in the i 

 orbit produced, and therefore, by (189), there is no disturbing force 

 perpendicular to the moon's orbit. 



(200.) Secondly : suppose the line of nodes to be in quadratures, or 

 to be perpendicular to the line drawn from the earth to the sun, as in 



Fig. IS. 



f'j. 43. The sun, in the figure, may be considered as being below the 

 plane of the moon's orbit. Also, the moon's.' m the earth 



being small, the points, at which the moon's distance from the sun ix 

 the same a* the earth'*, are very nearly the same as tl 

 quadrature, or (in the case before us) they are very nearly the same as 



