154 THE TIDAL PROBLEM. 



In this case the lower limit of the mass of the system can not be computed 

 except an assumption be made regarding the relation between tw^ and Wj. 



Suppose — -=ti- Then equation (16) gives 



m, + m,= (l+/.)m,^^^^i^^ (17) 



When P is expressed in mean solar days, and v^ in kilometers per second, we 

 have , «j^ 



^n,■Y1n,^^^Pv,'{l-\-|xy (18) 



where mi-\-m2 is expressed in terms of the sun's mass. If it were true that 

 the more massive star always gives the observed spectrum we should have 

 /i>land ,424 



^i + rn,>^Pv,' (19) 



Let Oj and Oj represent the radii of Wj and Wj respectively. Let the dis- 

 tance between their surfaces be represented by ku. Then 0^k<1 and we 

 have o / I N-! 



ai + a2 = a(l— k) F = , (20) 



fc(l — K)K/mj + W2 



Suppose the two bodies have the same density o. Then we have from 



Tw, = i^Traa,', Wj = -K^7raa/ and equation (20), putting — ^ = /( as before 



3;r (1+;^^)^ ^ 



A;2P='(l-/c)3(l+//) ^ ^ 



The ratio /£ may vary from zero to infinity, and k from to 1. From the 



derivative ^ o /1 1 '\2/i ax 



da _ St: (1+^)^1 ~i"^) 



d^~A;2p2(i_^)3 ^§(l+//)2 



it follows that, for fixed values of P and «, (t constantly increases wliile fx 

 varies from to 1, and then constantly decreases while /x varies from 1 to 

 00 . Therefore, since (21) is a reciprocal equation in //, we have 



12^ ^^^ Stt^ (22) 



k\i-Kyp^^ '=k\l-Kyp- 

 OT, changing the units so that a will be expressed in terms of the density of 

 water when P is expressed in mean solar days, we have 



100(l-/c)3P^-''-100(l-«)3p2 ^^^^ 



The smaller k the smaller the limits for o. When the bodies are in contact, 

 and the period 4.57 hours, that of ^ Cephei,' we find 



2.2><7>0.5 



The Period of /3 Cephei, by E. B. Frost, Astrophysical Jour., 24 (1906), pp. 259-262. 



