Oct. I, 1874] 



NA TURE 



451 



improbable solution of that part of the secular inequality in the 

 moon's mean motion which remains still unex])lained. 

 ■ He pointed out that inasmuch as the axis of the tidal spheroid 

 is always behind the moon's place, a couple is exerted by the 

 forces of the moon's attraction, which on the one hand retards 

 the rotation of the earth, and on the other increases the dimen- 

 sions of the lunar orbit. 



This alteration of the lunar orbit prevents us from concluding, 

 as we should otherwise do, that the kinetic energy which passes 

 into heat in the movement of the tides has for its exact equivalent 

 a corresponding quantity drawn from the store laid up in the 

 earth's rotation on its axis. 



The object of the present communication is to examine whether 

 we can assert such an equivalence to hold approximately, and if 

 so, to what degree of approximation. The question was started 

 some years agi) by the Astronomer Koyal in the Astronomical 

 Notices for tlie year 1S66. 



It occurred to the author that we might arrive at a solution of 

 the problem from the information given us by the equation of 

 energy combined with that of the conservation of angular 

 momentum. 



Let us in the first place take the case of a binary system con- 

 sisting of the earth and moon, but suppose the plane of the 

 earth's equator to coincide with that of the lunar orbit. If Q 

 denote the energy which, during a given interval, passes into 

 heat through tidal action, then, assuming the moon spherical 

 and her rotation consequently unaltered, (1 = - 5 (energy of 

 earth's rotation) - S (energy of lunar orbit). By the energy of 

 the lunar orbit is denoted the kinetic energy of the revolution 

 of the earth and moon round their common centre of gravity, 

 together with the potential energy of their separation. 



Now the energy of orbit = constant - J m m^ ji — , where 



a 

 til tii'^ represent the masses of the two bodies, fi the unit of attrac- 

 tive force, and a the mean distance. 



Hence Q = - 5 (energy of earth's rotation) - J m in''- jn —^. 



Let // denote the angular momentum of the revolution of the 

 two bodies round their common centre of gravity, // the angular 

 momentum of the earth's rotation, then 

 5 // = - 8 /i 



but 



III m^ Vm ,_ 



A.y^*;=^i V « VI - e^ 



When the excentricity is small the second term in this expression 

 maybe shown to be negligible when compared with the first, and 

 we may write 



6//= -S/.= - '".'"^ift -^ 



.-.()=_ 5 (energy of tarth's rotation) + ^" + «^^ g ^y 



Or if / denote the moment of inertia of the earth round her 

 axis, 

 10 her angular velocity of rotation, 

 n the mean angular velocity of the moon in her orbit, 

 e^ =: - /w5w + /nSw 

 • _ /„8„ = ^L^ 



The left-hand member represents the loss of energy due to the 

 slackening of the earth's rotation, and as Jl has the same sign as 

 cu, we learn that not only is all the energy Q which is turned into 

 heat in the motion of the tides drawn IVom the earth's rotation, 

 but that, as a necessary concomitant, additional energy is trans- 

 ferred from the earth's rotation to the store at potential and 

 actual energy, corresponding to the orbital motion of the system. 



It also follows that when n is small compared to w [in the 



actual case — rx — nearly], the energy so transferred bears a 



u 27 

 very small ratio to Q, and that the energy lost in the earth's rota- 

 tion is almost the exact equivalent of that consumed in tidal 

 friction. 



Let us now consider the case which we actually have to deal 



with, where the plane of the earth's equator does not coincide 

 with the plane of the orbit. 



Let G represent the resultant angular momentum of the 

 system which will l)e2fixed in magnitude and in direction. 



$, the angles which the planes of A and // make with the 

 plane of G. 



Then, since //- = G- + /i- - 2 G /i cos e 



//s //■= {// - G cos e) s/i + Gh sin e 5 e 



II, 11= ^'ijl^i - G cos,a)^+ G,U^^,,,\ 



Or, 8 // : 



m »/' ij II. ^ a 



cos(0 -1- e) if -f sin (0 + e) 8 e] 



The author proves from a calculation of the" disturbing reac- 

 tionary forces exercised by the tidal protuberances that the 



variations 5 Sand - are of the same order of magnitude, although 



their exact ratio cannot be determined without far more com- 

 plete data respecting the tides than we at present possess. 



Let the ratio of the first of these variations to the second be 

 denoted by \, then 



m m^ rJiL ( \ Sa 



8H = - ,— - . <i-Atan (0 + 



Ia.8. 





jz'^a 



sec (0" -1- I 

 ^^^(tan0 +1) I 



We may therefore still infer that since n is small compared 

 to a, the energy lost in the earth's rotation is almost the exact 

 equivalent of that consumed in tidal friction. 



The same conclusion manifestly applies to the work done by a 

 tide-mill or any other mechanism in which the tides furnish the 

 motive power. 



It would further appear that as the mean value of tan (0-re) 

 is less than 4, and that of X cannot, on any probable hypothesis 

 of the position of the tides, be supposed to exceed unity, the co- 



n 



efficient of — in the above expression is positive. Hence we 



may conclude that, as in the simpler case previously discussed, 

 the small transfer of energy which accompanies the principal 

 action takes place from the earth's rotation to the moon's orbit. 



All these conclusions apply inuialis iiiutamiis if we regard as 

 our binary system the earth and sun. 



In the case of nature, where we have to consider the three bodies 

 acting together, the main conclusion that aU the energy lost in 

 tidal friction is drawn from the earth's rotation will not be in- 

 validated. 



Moreover, if we assume, as is generally done, that the fric- 

 tion varies as the velocity, the lesser effect, i.e. tlie concomitant, 

 transfers its energy from the earth's rotation to the energy of the 

 orbit of the moon about the earth, and that of the earth about the 

 sun will correspond to the values separately calculated for the 

 binary systems. 



On the construction of large NicoFs Prisms, by W. Ladd. — 

 In January 1869 I constructed two Nicol's prisms of about 2^ in. 

 aperture, which in the able hands of Mr. W. Spottiswoode and 

 Dr. Tyndall have done much valuable work, and given rise to a 

 great demand for such prisms, both in England and America ; 

 but as the length of a good Nicol should be about three times 

 its diameter, very great difficulty is experienced in procuring 

 pieces of spar of sufficient purity to give such a field. 



This has given rise to various methods of utilising the spar by 

 building up prisms of shorter pieces and combining them in such a 

 way as to unite their field of view, such as utilising four prisms 

 of I in. aperture, thus giving an aperture of 2 in. Another plan 

 I adopted was to unite two whose diameter in one direction was 

 double that of the other ; these, being balsanitd together, made a 

 very good prism ; but lately I had a very good piece of spar that, 

 but for one corner of the rhombus, which was bad, would have 

 made a prism 3] in. aperture. This was, therefore, too valuable 

 a piece to be put aside. 



I therefore cut it at the proper angle, which took away all the 

 bad portion ; I then took another piece half the length of the 

 first, but of the same diameter, and cut this also at the proper 

 angle, and the bringing the two ends together gave me another 

 complete half ; these, having been balsamed together and united 

 with the first half, produced a perfectly good prism. I may add 

 that it is essential that the two or more pieces constituting the 

 half prism should have tlieir cleavage planes exactly parallel, or 

 the image would be bent at their junction. 



