Sept. 23, 1875] 



NATURE 



455 



acceptance which his endeavour to solva the climatal problems 

 of past epochs by astronomical computation has very deservedly 

 met with on the part of Geologists, his denial of the possibility 

 of a thermal circulation in the ocean is utterly repudiated alike 

 on mathematical and on experiential grounds, by those whose 

 authority as physicists ought to make him feel less confident in 

 his own conception of the question. W. B. Carpenter 



Source of VoIcanic^Energy 



A FEW words of explanation are necessary "concerning my 

 letter which appeared in Nature, vol. xii. p. 396. Mr. Mallet's 

 prime source of energy for producing tangential pressures is the 

 force of cohesion developed in a cooling globe, gravitation giving 

 only partial assistance ; and when I spoke of "gravitation of the 

 whole mass to itself," I wished to convey that, setting aside alto- 

 gether the force of cohesion and its accompanying motions, there 

 still remains the force of gravitation, which, acting in a globe of 

 such size as the earth, and composed of heterogeneous materials, 

 must of itself produce enormous local pressures. 



Mr. Fisher objects to my supposing the possibility of the de- 

 velopment of heat without room being left for motion, but so far 

 as I understand the doctrine of energy, it is only necessary to 

 Xr&y^ force for the production of heat when motion is impossible. 



In Mr. Fisher's interesting paper his objection appears to be 

 to the localisation oi fusing, and not to the localisation of heat, 

 fusing in some cases being prevented by the accompanying pres- 

 sure. But in my little diagram I attempted to explain that the 

 forces producing the high temperature might act in one set of 

 strata, the neighbouring strata above and below at the same time 

 being under much lower pressure, the pressure upon them being 

 equal to the pressure of the rocks doing the work, minus the 

 cohesion of said rocks ; this difference of pressure being sufficient 

 to allow one set of rocks to melt while others are crushed. 



Kenmare, Co. Kerry Wm. S. Green 



Gyrostat Problem : Spinning-top Problem 

 In vol. xi. p. 424 is given the solution, by Sir W. Thomson, of 

 his gyrostat problem at p. 385. I venture to send a slightly dif- 

 ferent method * of obtaining the retult (far inferior to Sir W. 

 Thomson's in elegance and simplicity), in which Euler's equa- 

 tions for the motion of a rigid body about a fixed point are 

 employed. 



I . Take point of suspension for origin ; the string for axis of 

 s. The axis of the wheel oxf revolves in horizontal plane xoy 

 with uniform angular velocity n, and the wheel revolves round 

 its axis Ojt' with angular velocity w-^. The weight of wheel and 

 axis will have moments round an axis oy in horizontal plane 



perpendicular to Ox'. Let w' = weight of wheel and axis ; 

 A, B, B, moments of inertia round oz, ox', oy' ; w'2 angular 

 velocity round oy' at time t ; a "= the distance of c. g. from oz, 

 xox! = <p = angle described by ox' in time /. Taking moments 

 about oy, we have 



B d a', 



Yt + A-Bw^Cl^v/ag ... (I) 



(Pratt, " Mech. Phil." 446). Also since there is no velocity 



* A comparison of this method with Sir W. Thomson's (which is virtually 

 the same as that adopted by Airy in his tract on Precession and Nutation) 

 is instructive as illustrating' the dynamiod meaning of Euler's equationi.— 

 Ed. Nature. 



about an axis in horizontal'plane perpendicular to resultant axis 



of Wi Wo, 



where 



Wi sm. f - Wj cos. =■ o 

 <p = at. 



(2) 



.•. -^j = '^^i fl sec. V = 7(/i n for ^ = o in (i), since w^, n are 

 independent'of the time ; whence (1) becomes 



A w^ n. = TV a g, 

 where A = wk^,'n = i . . . . q.e.d. 



2. A similar question (concerning a spinning top) was proposed 

 in the Senate House, Cambridge, in 1859, of which indeed th« 

 preceding is a particular case. 



A uniform top spins upon a perfectly rough horizontal plane, 

 its axis being inclined to the vertical at a constant angle a, and 

 revolving about it with constant angular velocity n. Prove that 

 the velocity of rotation of the top about its axis must bs 

 (a^ + -*')n- cos. a + p-fl , •,,,.,. 

 '~r'» « > '^^^^^ a is the distance of the centr* 



of gravity from the extremity of the peg, k' k the radii of gyration 

 about the axis of figure, and about an axis through c. o. perpen- 

 dicular to it respectively. Take o, the extremity of the peg, 

 which remains fixed, as origin, and let o z' be position of axis at 

 any time / ; o G = a ; z o z' = o. Let M - mass of the top ; 

 A, C, C, moments of inertia about ox', oy, o z' (rectangular 

 axes moving with the top) ; iv^ w^ w^, angular velocities about 

 o x, oy', o z' at time t. 



The intersection of planes xoy, .jr'oy will move round o z 

 with angular velocity fl. Let <p = angle which o x' makes with 

 this line. 



If we take moments about o x', we have by Euler's equations 

 (Pratt, art. 446) — 



A dw, — 



•— TT- -)- C — Azu^w^ — Mgaco%.zy^ . . (i) 



Also 7^1 



d<b 

 n sin. <(> sm. a, Wj = n cos. <p sm. a, w, = — - + n cos. a 



d t 

 cos. zy^ — cos. <p sin. a (ibid. 447) ; 



.". -^ = n cos. <p sin. o -/ = n Cos. * sin. o [w. - n cos. o). 

 dt dt T \ » I 



Substituting in''(i) and reducing, we get — 



C n W3 = Mg a + A a- cos. o. . . . (2) 

 But A = Af(P + a% C =:^V; 



• w - .g^ + (^' + ^'') Q' cos, g 



If a = . 90" in equation (2), we'get^the solution of the preceding 

 question as a particular case. F. M. S. 



Arnesby 



OUR ASTRONOMICAL COLUMN 



The Mass of Jupiter.— M. Leverrier has made a 

 special communication to the Paris Academy of Sciences 

 with reference to the bearing of his researches on the 

 motion of Saturn, in a period of 120 years, on the value 

 of Jupiter's mass. Laplace, in the Mdcanique Celeste^ 



had fixed — ^ making use of the elongation of the 



1067-09 

 fourth satellite as determined by the observations of 

 Pound, the contemporary of Newton, observations of 

 which it appears we have no knowledge, except from 

 the reference to them in the " Principia ; " subsequently 

 Bouvard, comparing Laplace's formulae with a great 

 number of observations, discussed with particular care, 

 constructed new Tables of Jupiter, Saturn, and Ura- 

 nus, in which important work he formed equations 

 of condition, wherein the masses of the planets entered 

 as indetcrminates, and by the solution of which their 

 values adopted in the Tables were obtained. The 

 denominator for Jupiter's mass, expressed as a fraction of 

 the sun's taken as unity, is 10700, and Laplace stated that 

 on applying his theory of probabilities to Bouvard's 

 equations it appeared to be nearly a million to one 

 against the error of the mass thus deduced, amounting to 

 one- hundredth part of the whole. M. Leverrier then 



