34 



KNOWLEDGE. 



[Februaby, 1908. 



We van now apjilv the formula found above, viz. — 



" PM = If 

 PD, 



to tiud the diameter of the path of the Moon at I'ull, 



treated as jiart of a circle. 



For, clearly. PM — resultant fall towards Sun in an 



hour =- ol-823 ; also PR = resultant velocity per hour 



= 68864-46. 



(68864-46;= 



34-823 



miles. 



Whence PD = diameter of path — - 

 And the radius of path is half this, or 68,091,700 miles. 



But the Sun at this distance makes the Earth fall 

 23-859 miles in an hour. And since the distances have 

 been made the same, the falls are jiroportional to the 

 masses of the attracting bodies. 

 Whence 



Mass of Sun _ 23-859 ,, f 9-2,885,400 | ■' 



\ 238,? 



10-964 



1,818 



Mass of Earth + Moon 

 = 329,200. 



The other result is to find at what distance from the 

 Earth the Moon -would have to be for her jjath round the 

 Sun to be just straight (neither concave nor convex) at 



THE PATHS OF JUPITER'S MOONS 

 DRAWM B\ C T VVHITMEU- 



^>.^^^., 



"^^ 



O/- 



■^O/v 



^'«^^-. ^ 



^^- 



.fiC.X. 



<f^- 



= 160,252,500 miles. 



In a similar manner we find that the radius of the path 

 of the Moon at New, treated as part of a circle, is 

 (64287-70) » __ 

 2x12 895 



It is fairlv evident that these are the regions of 

 maximum and minimum concavity, and that the radius of 

 the path -will gradually increase as we pass from Full 

 Moon to New. 



Before passing on from these figures, we shall indicate 

 two other interesting results connected with them. 



First we can obtain the mass of the Sun comjiared with 

 the combined mass of the Earth and Moon. Suppose 

 the Moon were moved to a distance from the Earth equal 

 to the Sun's distance, then her fall towards the Earth in 

 an hour would be 



(238,818 ") 

 ( 92385,400 1 

 since it is inversely proportional to the square of the 

 distance. 



10-964 



miles 



New Moon (type (2) given above). 



This will clearly be the case if the fall of the Moon 

 towards the Earth in one hour is exactly equal to the fall 

 of the Earth towards the Sun ; i.e., we must make the 

 former fall equal to 23859 miles. But this fall varies 

 inversely as the square of the distance. 



jEequn-ed distance") -' 10964 



Hence 



238,818 



i 



23-859 



Whence the reipiired distance = 161,889 miles. 



This distance is lilcewise the radius of the sjihere over 

 w hich the Earth holds supreme sway ; at its boundary the 

 attractions of the Earth and Sun are equal, while within 

 it the Earth's attraction exceeds that of any other body. 

 All satelUtes, except our Moon, lie within the sphere over 

 which their Primary rules supreme ; this is the reason 

 why their paths in all cases belong to types (3), (4), 

 or (5) given above. 



The reader acquainted with the geometry of curves of 



