442 PHYSICAL SCIENCES. 



tend to move in the orbit p' p" n", whose plane cuts the ecliptic in the 

 straight line S n". The action of the distui'bing force on the planet when 

 at p" will bring the node to n'", and so on. In this manner the node goes 

 backwards through the successive points n, n', n", n'", &c., and the line of 

 nodes S n has a perpetual retrograde motion about S, the centre of the sun. 

 The disturbing force has been represented as acting at intervals for the sake 

 of illustration : in nature it is continuous, so that the motion of the node 

 is continuous also ; though it is sometimes rapid and sometimes slow, 

 now retrograde and now direct ; but, on the whole, the motion is slowly 

 retrograde. 



NOTE 74, p. 18. When the disturbing planet is anywhere in the line 

 S X, fig. 19, or in its prolongation, it is in the same plane with the dis- 

 turbed planet ; and, however much it may affect its motions in that plane, 

 it can have no tendency to draw it out of it. But when the disturbing 

 planet is in P, at right angles to the line S N, and not in the plane of the 

 orbit, it has a powerful effect on the motion of the nodes : between these 

 two positions there is great variety of action. 



NOTE 75, p. 19. The changes in the inclination are extremely minute 

 when compared with the motion of the node, as evidently appears from 

 fig. 19, where the angles np n', n' p' n", &c., are much smaller than the 

 corresponding angles n S n', S n", &c. 



NOTE 76, p. 20. Sines and cosines. Figure 4 is a circle ; np is the 

 sine, and Cp is the cosine of an arc mn. Suppose the radius C m to 

 begin to revolve at m, in the direction mna; then at the point m the 

 sine is zero, and the cosine is equal to the radius C m. As the line C m 

 revolves and takes the successive positions Cn, C a, Cb, &c., the sines 

 np, aq, br, &c., of the arcs mn, ma, mh, &c., increase, while the 

 corresponding cosines Cp, Cq, Cr, &c., decrease ; and when the revolving 

 radius takes the position C d, at right angles to the diameter g m, the sine 

 becomes equal to the radius C d, and the cosine is zero. After passing the 

 point d, the contrary happens; for the sines e K, IV, &c., diminish, and 

 the cosines C K, C V, &c., go on increasing, till at g the sine is zero, and 

 the cosine is equal to the radius C g. The same alternation takes place 

 through the remaining parts gh,hm, of the circle, so that a sine or cosine 

 never can exceed the radius. As the rotation of the earth is invariable, 

 each point of its surface passes through a complete circle, or 360 degrees, 

 in twenty-four hours, at a rate of 15 degrees in an hour. Time, there- 

 fore, becomes a measure of angular motion, and vice versa, the arcs of a 

 circle a measure of time, since these two quantities vary simultaneously 

 and equably ; and, as the sines and cosines of the arcs are expressed in 

 terms of the time, they vary with it. Therefore, however long the time 

 may be, and how often soever the radius may revolve round the circle, 

 the sines and cosines never can exceed the radius ; and, as the radius is 

 assumed to be equal to unity, their values oscillate between unity 

 and zero. 



NOTE 77, p. 20. The small excentricities and inclinations of the 

 planetary orbits, and the revolutions of all the bodies in the same direction, 

 were proved by Euler, La Grange, and La Place, to be conditions neces- 

 sary for the stability of the solar system. Subsequently, however, the 

 periodicity of the terms of the series expressing the perturbations was 



