130, i3i] The Tidal Problem 131 



quickly for the series of equilibrium configurations discussed in 122 to be 

 able to keep pace with its motion. 



There will be a lag in the orientation of the primary so that its major- 

 axis may be expected to point to some position such as T* in fig. 24, when 

 the tide-raising mass is actually in 

 a position such as T at a distance / 

 ahead of T '. 



The potential of the tide-gene- 

 rating mass at T' would be fl = M'jr 

 where r is the distance OT', but the 

 true potential produced by the mass 

 actually at T is Fig 



' 





The correction which has to be introduced is that arising from the last 

 term. For simplicity the primary may be regarded as a chain of matter 

 lying along the axis of a?, and the effect of the correction is to introduce 

 a force of amount M'l/r 3 per unit mass perpendicular to the axis of a\ 



Replacing r by its value R x, we find 

 M'l M'l 



The first term will produce a uniform acceleration M'ljR* along the axis 

 of y. Combining this with the acceleration M'jR? towards T which the 

 primary has so far been supposed to have, we obtain a resultant acceleration 

 AT/R* towards T. 



The remaining terms on the right of equation (385) set up various 

 distortions. The second term sets up a uniform rotation at a rate 3M'l/R* 

 per unit time ; the third twists the major-axis of the primary into a piece of 

 a parabola, the next superposes a cubical distortion, and so on. It can be 

 readily seen from equation (385) that the combined effect of all these dis- 

 tortions will be to set up such a motion that initially the axis of the spheroid 

 is bent to the shape of a piece of the curve y = I/a? 3 , a curve shaped some- 

 what like a boomerang; there seems to be no tendency for the axis of the 

 primary to assume the shape of a logarithmic spiral, which is the observed 

 shape of the spiral nebulae. 



This last result has an obvious bearing on the tenability of the " Planet- 

 esimal Theory" of Chamberlin and Moulton, described in 15. 



92 



