February, 1903.] 



KNOWLEDGE. 



33 



round the Sun by supposing the man to describe a circle 

 of fourteen feet radius in fifty-nine seconds, while the ship 

 is describing a circle of one mile radius about a light- 

 house, or other fixed point in the sea, in twelve minutes 

 ten seconds. Then the man's path an the deck will corre- 

 spond to the Moon's path round tlie Earth, and his path 

 on the sea to that round the Sun. Two spconds in the 

 model correspond to one day in nature, while one mile 

 corresponds to 93,000,000 miles. 



We shall first discuss the nature of the resultant path 

 on the assumption that the orbits of the Moon round the 

 Earth and of the Earth round the Sun are circles in the 



Fio. 1. — Tlie Curvature of the Moon's rath 



same plane described with uniform speed, and then pass 

 on to a more accurate description of the nature of the 

 Moon's path round the Earth, indicating a few of the many 

 different ways in which it differs from uniform circular 

 motion. 



There is a whole class of curves, known as trochoids, 

 described by points that move uniformly in a circle, whose 

 centre moves uniformly in another circle in the same 

 plane. As all the motions of rotation and revolution in 

 tlie planetary system (excepting the satellites and pro- 

 bably the rotations of Uranus and Neptune) take place in 

 the same direction, we shall only consider the case where 

 the motion in the two circles is in the same direction. 

 The resulting curves may be divided into the following 

 five groups : — 



(1) Those that are wholly concave to tlie centre of the 

 larger circle. 



(:2) Those that are just straight at points corresponding 

 to New Moon and concave elsewhere. 



(3) Those that are partly concave and partly convex to 

 the centre of the larger circle, the two regions being 

 sopai-ated by "^loints of inflexion" where the curve is 

 straight for a small space. 



(4) Those where the points of intlexiou coalesce in pairs 

 and form cusps. 



(5) Looped paths ; these occur when the linear velocity 

 of the i>oint in the small circle exceeds that of its centre 

 in the largo circle. 



Representatives of all these types, except the second, 

 are to be found in the solar system. 



Our Moon is the solitary example of the first type, the 

 only case in which the Sun's attraction exceeds the attrac- 

 tion of the primary. 



I'iiobos, Deimos, Ganymede, Callisto, Kliea, Titan. 

 Hyperion, and Japetus, belong to typo (3). 



Europa and Dione at certain times conform exactly to 

 type (4), and come to rest at New Moon. At other 

 times they belong alternately to (3) or (.i). 



lo, Jupiter's Satellite V, Tethys, Enceladus, and Mimas 

 belong to type (5). 



The circumstances of Jupiter's satellites have been 

 investigated by Mr. C. T Whitmell in the Journal of 

 the Leeds Astronomical Society, 1901, page 98. His 

 diagram is reproduced by permission from that paper. 

 The path of II. Europa, when it conforms exactly to type 

 (4), is an epicycloid, or figure traced by a point on the 

 rim of a wheel which rolls without slipping round the 

 outside of a fixed wheel. When the fixed wheel is of 

 infinite size, that is when it becomes a straight line, the 

 epicycloid becomes a cycloid ; and this is practically the 

 case with Europa, since the path of Jupiter r<3und the 

 Sun is so enormous compared with the path of Europa 

 round Jupiter. It only differs from curve II. on the 

 diagram by having a cusp at N instead of the small looj". 



It is not difficult to prove by elementary methods that 

 the path of the Moon is everywliere concave to the Sun. 

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

 towards the Sun in a given time (say one hour) exceeds 

 the fall of the Moon towards the Earth in the same time. 

 The word " fall " here means the amount by which the 

 body is bent away from the tangent to its path by the 

 action of the attracting body in the centre. Thus, let 

 QPRD be the Moon's circular path round the Earth, C, in 

 the centre ; P, the Moon's position at any selected time, 

 and PT the tangent to her path at this time. QR are 

 her positions one hour earlier and one hour later, the 

 distance QP, PR being greatly exaggerated for clearness. 



Then, clearly, if QR cuts PC in M, QM = RM, and the 

 fall of the Moon in one hour = TR = PM. 



Now by Euclid (III, 3-5) PM.MD = RM.MQ = RM^. 

 Now since the distance PR is such a small jmrt of the 

 circumference, RM is sensibly equal to PR, and MD is 

 sensibly equal to PD. Hence we have PM.PD — PR-, 



Now PD = 238,818 x 2 miles. 



Also the circumference = PD x 3141(j and the time 

 taken to describe circumference = 273216 x 24 hours, 

 circumference PD x 31416 



Whence PR = 



And PM = PD X 



27-3216 X 24 27 3216 x 24 



1 3-1 416 y- 



\ 27-3216 X 24 1 ■ 



by logarithms we find that PM = 



Working this out 

 10-9G4 miles. 



We can find the fall of the Earth towards the Sun in 

 one hour by exactly similar reasoning.' 



PD is now equal to 92,885,400 x 2 miles ; also instead 

 of 27-3216 we now write 365-2564, being the number of 

 days in the year. 



Whence fall of Earth towards Sun in one hour 



= 92.885.400 - * ^'^"^^'^ '' 



= 23-859 miles. 



365-2564 x 24 



We see from this that the fall of the Earth towards the 

 Sun in one hour is considerably more than double that 

 of the Moon towards the Earth. It follows that the 

 resultant fall of tlie Aloon is always towards the Sun, and 

 consequently that her jjath is everywhere concave towai-ds 

 him. Tiie concavity is, however, very much greater at 

 Full INloon than at New; for in the first case the total fall 

 towards the Sun in an hour is 23-859 -f 10-964 miles 

 = 31-823; in the second case it is 23 859 — 10-964 

 = 12-895, or little more than one-third of the first result. 



The resultant velocity iu miles per hour is 



66576-08 -t-'2288-38 = 68864-46 at Full Moon. 

 66576-08 - 2288-38 = (54287-70 at Now Moon. 



