170 



Mr. G. H. Darwin. 



[June 19, 



It may be well to notice that x is proportional to the square root of 

 the satellite's distance from the planet. 

 Then the equations (1) and (2) become 



h=y+x . (3), 



X" X" 



(3) is the equation of conservation of moment of momentum, or 

 shortly, the equation of momentum ; (4) is the equation of energy. 



Now, consider a system started with given positive (or say clock- 

 wise) moment of momentum h ; we have all sorts of ways in which it 

 may be started. If the two rotations be of opposite kinds, it is clear 

 that we may start the system with any amount of energy however 

 great, but the true maxima and minima of energy compatible with the 



given moment of momentum are given by —=0, 



dx 



or x — h + 2-=0, 



■x 6 



or a3 4 -7^ 3 + l = . (5). 



We shall presently see that this biquadratic has either two real 

 roots and two imaginary, or all imaginary roots. 



This biquadratic may be derived from quite a different considera- 

 tion, viz., by finding the condition under which the satellite may 

 move round the planet, so that the planet shall always show the same 

 face to the satellite, in fact, so that they move as parts of one rigid 

 body. 



The condition is simply that the satellite's orbital angular velocity 



jtxl— (M+m) =1 + ^\ {—) 0- + v ) } =-, suppose 



in the case of the earth, considered as heterogeneous, the -§ would be replaced by 

 about 



It is clear that si is a time ; and in the case of the earth and moon (with v = 82), 



sf=3 hrs. 41 niins., if the earth be homogeneous, and 



si = 2 hrs. 41 mins. if the earth be heterogeneous. 



For the units of length and mass we have only to choose them so that f Md 2 , or - 

 \Ma 2 , may be unity. 



With these units it will be found that for the present length of day »='8056 

 (homog.) or '7026 (heterog.), and that 



A = -8056[l + 4-01] =4-03 (homog.), 

 or &=-7026[l + 4-38] =378 (heterog.) 



For the value 4*38 see Thomson and Tait's u Nat. Phil.," § 2/6, where tidal fric- 

 tion is considered. 



