RESULTING FROM THE THEORY OF THE GYROSCOPE. 13 



If we multiply tlic coefficient of sin nj in (44) by sin /, we shall have the \alue 

 as an arc of a great circle of this fluctuating displacement called tlie equation of the 

 equinoxes in longitude. The coefficient thus modified will represent the minor semi- 

 diameter, and that of the nutation proper (45), the major semi-diameter of the 

 ellipse of nutation ; they have the ratio cos 2/ : cos / nearly. This ellipse has its 

 major axis (equal to about 18" of arc) directed towards the pole of the ecliptic. 



The period of its description — is that of the revolution of the moon's nodes. At 



the same time a minute ellipse of semi-annual nutation due to the sun is super- 

 imposed upon this. It has its longer axis (about one second of arc) likewise directed 

 to the pole of the ecliptic. The smaller axis is to the major as cos /: 1. 



It is easy to show that the precession caused by tlie sun and moon is equal (with 

 slight difference due to the ratio we have just been considering) to what it would be 

 if those bodies were uniformly distributed in solid rings over circles (in the plane 

 of the ecliptic) about the centre of the earth, having radii equal to their mean dis- 

 tances. In this case, unless there was a particular relation between the couple 

 producing- the initial rotation of the earth and that arising from the attraction of 

 the two rings, there would be an extremely minute nutation, of which the period 



would be (see equation 9) -,-, = T^v'y '^vhich is almost identical with the siderial 



2k 11 C 



the ratio being ., • 1- If we suppose the primitive rotation of the earth 



n ' '-" 



to be that alone, about its axis of figure n, then the nutation will exhibit tlie com- 

 mon cycloidal motion of equation (10); but its total amount would be but about 

 Jf second of arc. 



5 5 



An explanation of the deviation of rifled projectiles will be found in what is 

 said (pages 8 and 9) in reference to the conversion of gyration about a shifting 

 axis into precession about an axis perpendicular to the plane of motion of the 

 first. Elongated rifled projectiles, while they maintain almost unaltered their 

 " angle of elevation," are found to deviate with great uniformity from the vertical 

 plane of projection, in a direction corresponding to the twist of the gun, while 

 spherical projectiles (fired from rifled guns) having precisely the same rotary 

 motion, do not so deviate ; showing that the cause of the phenomenon is something 

 else than the direct action of friction or pressure of the air. 



In the remarks made in the paragraphs just referred to, it is explained how 

 the consecutive small annual gyrations about the line from the centre of the earth 

 to the sun (in the tropics) become, in their integral, the movement which we call 

 precession, about an axis perpendicular to the plane of motion of that line, inasmuch 

 as each small primary gyration causes a corresponding shifting of the line about 

 which it takes place. Something very similar occurs to produce the deviation of 

 elongated projectiles. In issuing from the gun, the resultant of the atmospheric 

 resistance (denoted by the arrow R) coincides with the axis of the projectile, and it 

 has no other effect than to retard the motion of translation ; but the action of 

 gravity caiises the trajectory to curve downwards, and the direction of the atmo- 

 spheric resistance becomes oblique to the axis of the projectile (at «'), and (in 



/27l\ 



aay ( — ), 



