82 



KNOWLEDGE 



[May 2, 1892. 



by the attraction of the other will lag, and if the viscosity 

 is small the angle of the lag will be only a few degrees. 

 For simplicity we shall now treat the spheroid Sol as 

 having its mass collected at its centre of gravity, and 

 examine the effects on the eccentricity arising from the 

 tidal reaction of Helios ; but it must be remembered 

 that in general the whole effect of tidal friction in the 

 system of stars will depend upon the aggregate effect of 

 the double tidal reaction arising from the rotations of both 

 bodies — a complication that renders the rigorous investi- 

 gation in general very diftionlt. 



With Sol thus reduced to a weigjjied point revolving in 

 the plane of the equator and raising tides iu Helios, the 

 tidal configuration will be something like that indicated in 

 Fiff. 1. 



fSol) 



Fig. 1. 



In the position of the tidal ellipsoid of Helios shown 

 in tlie figure the whole attraction on Sol does not 

 pass through the centre of inertia V (about which 

 Helios rotates), but through some point i-. The reaction 

 of Sol is equal and opposite, and hence there arises a 

 couple (with arm c C) acting against the rotation of Helios. 

 We may resolve the whole attraction of Helios (c' c) into 

 two components, one of which (r' C) passes through the 

 centre of inertia C and produces no effect, as it is counter- 

 balanced by the centrifugal force of the revolving body. 

 The other component [c' d') perpendicular to the radius 

 vector is unbalanced by any opposite force, and hence 

 acting as an accelerating force tends to increase the in- 

 stantaneous linear velocity, whereby there results an 

 increase in Sol's mean distance. 



As the axial rotation of Helios is reduced, Sol is wound 

 off on a spiral whose coils are coincident and very close 

 together. To speak mathematically, the iiKniioit nf 

 momcntniii of the whole system is nm»U(nt,'-'- and since the 

 reduction of Helios' rotation causes the axial moment of 

 momentum to diminish, it follows that the moment of 

 momentum of orbital motion must augment. In other 

 words, tidal friction transfers moment of momentum of 

 axial rotation to raoment of momentum of orbital motion, 

 and hence the mean distance must increase. 



With these very brief introductory remarks, let us now 

 e.xamine the changes of the eccentricity of the orbit. In 

 the mathematical works on the tidal theory it is shown 

 that the tide-generating force varies inversely as the cube 

 of the distance of the tide-raising body. The height of 

 the tide, according to the principle of oscillations, varies 

 as the square of the tide generating force, or inversely as 

 the sixth power of the distance. From Fig. 1 it is easy to 

 see that (for a given lag and given height of the tide) the 



* The energy of the system, however, is not constniit, hut con- 

 tinually diminishing, owing to loss of radiant energy. 



Fig. 2. 



tangential force varies inversely as the distance. t There- 

 fore the tangential disturbing force varies inversely as the 

 seventh power of the distance of the. tide-raising body. 



When Sol is in Perihelion the t'des are higher (in the 

 inverse ratio of the sixth power of the distance) and the 

 tangential disturbing force is greater than when Sol is in 

 Aphelion, in the inverse ratio of the seventh power of the 

 Perihelion and Aphelion distances. It is well known in 

 the theories of planetary motion that a disturbing accelera- 

 tion at Perihelion causes the revolving body to swing out 

 further than it would otherwi.se have done, so that when 

 it comes round to Aphelion the distance is increased. 

 In like manner, an accelerating force at Aphelion increases 

 the Perihelion distance, somewhat as we have roughly 

 shown in Fig. 2. Now, if we consider the tidal frictional 

 component to act in- 

 stantaneously and only 

 at the apses of the 

 orbit, the effect would 

 be to increase the Peri- 

 helion as well as the 

 Aphelion distance, but 

 the latter at such an 

 abnormally rapid rate 

 that the orbit becomes 

 more eccentric. ; 



If the orbit is not 

 very eccentric similar 

 reasonmg to that just 

 employed for the two 

 apses could be applied to other opposite points in the orbit, 

 and the same general result would follow; when, however, 

 the eccentricity is considerable, this method of procedure 

 is not so satisfactory, though while the tides lag, as in Fig. 

 1, the eccentricity will continue to increase. 



We shall now present the effects of tidal friction as the 

 converse of those arising from a resisting medium, and 

 shall determine the law of the density of the medium 

 required to counteract the effects of tidal friction. Let us 

 consider the case in which the orbit has only a moderate 

 eccentricity (say not surpassing 0-3), since practically the 

 whole disturbing force due to the tides in Helios may then 

 be regarded as acting in the tangent to the orbit. When 

 the tides lag (less than 90°, as in Fig. 1), the tangential 

 component is directed forward, and hence tends to 

 accelerate the instantaneous linear velocity ; the force 

 arising from a resisting medium is directed continually 

 backward, and hence tends to cause the instantaneous 

 linear velocity to dimmish. The two forces are, therefore, 

 oppositely directed, and hence it is evident that if they 

 acted simultaneously the orbit would not undergo the 

 least change either in size or shape, but would be 

 rigorously stable. Now, the resistance encountered at any 

 given point of the orbit depends upon the density of the 

 medium, and is also proportional to the square of the 

 instantaneous linear velocity ; but from Kepler's law of 

 equal areas in equal times, it follows that the momentary 

 velocity of the revohdng body is inversely as the radius 

 vector. The accelerating force due to tidal friction varies 

 inversely as the seventh power of the distance ; therefore, 

 in order to counterbalance this by a retarding force due to 

 resistance we must suppose the density of the medium to 



t The tangential force is always equal to the whole f.irce acting in 

 the line c' c multiplied by sin t ; sin t, however (when r C is constant), 

 varies inversely as the radius (vector) p. 



X If the eccentricity is to remain constant the increase must be iu 

 the ratio of (1 — e) to (I + e) ; with tidal friction the ratio is more 

 nearly (1— e)' to (l + e)\ though not rigorously so, except when the 

 ecccntricitv is very small. 



