127-iso] The Tidal Problem 129 



and so may be regarded as the potential of a line density M'/v spread along 

 the path of the secondary. The whole value of 4> is accordingly equal to 

 the gravitational potential of a line density M'/v spread along the orbit of 

 the secondary. To an approximation we may neglect all parts of the orbit 

 except that near perihelion, so that the value of <$> will be~th"e potential 

 of a straight rod of line density M'jv, where v is the velocity at or near 

 perihelion. 



If R denote the distance at perihelion we readily find 



where a, y, z are rectangular coordinates having the centre of the primary 

 as origin, and the path of the secondary at perihelion is along the line 

 #=:.# 0) = 0. 



In this velocity potential the constant term does not affect the motion ; 

 the second term sets up a uniform velocity ^M'/R^v which does not alter the 

 configuration of the primary. 



The third term gives rise to impulsive velocities deducible from a velocity 

 potential 



(381). 



Now the impulsive velocities discussed in 123 were deducible from a 

 velocity potential (cf. equations (345) and (363)) 



-9* 







so that the two sets of impulsive velocities will agree if 



M' 



(382) ' 



The final term on the right of equation (380) indicates a tendency for 

 the axis c to shorten while the axis b lengthens, just as would happen if the 

 system were in rotation. There cannot ultimately be rotation in this case for 

 the tidal couples from the two halves of the orbit of the secondary exactly 

 neutralise one another ; it therefore appears that the values of b and c will 

 oscillate about the value b = c, as in 120, and under the influence of 

 viscosity the figure will ultimately resume its spheroidal form. 



Thus, neglecting terms in 1/^ 3 etc. it appears that the motion will be 

 the same, except for the preliminary oscillations in the values of b and c, as 

 j. c. 9 



