March 5, 1903] 



NA TURE 



415 



ties of the sediment, or mud, which I made, seemed to 

 indicate that the organic matter was condensed hydro- 

 carbon gases, or condensed volcanic vapours (such as one 

 might expect to be evolved unburnt in a very large volcanic 

 outburst). The sediment seems to be terrestrial, as the 

 large amount of organic matter, coupled with the small 

 amount of iron found, prohibits the theory of a meteoric 

 origin. 



The rain water contains 37'o grains of suspended matter, 

 or mud, to the gallon. 



The analysis of the suspended matter, dried at ioo° C, is 

 as follows : — 



Organic matter (loss on ignition) ... 364 per cent. 



Silica 45'6 ,, 



Alumina and oxide of iron... ... I3'6 ,, 



Magnesia ... ... ... ... 2'4 ,, 



Unclassified ... ... ... ... 2'o ,, 



Buckfastleigh, March 2. 



Rowland A. E irp. 



Proof of Lagrange's Equations of Motion, &c. 



In your issue of January 29, Mr. Heaviside put forward a 

 demonstration of Lagrange's equations of motion which appears 

 invalid. As neither his interpretation of Newton nor his 

 argument based thereon was stated with sufficient clearness to 

 enable a critic to locate the weak spot without running serious 

 risk of misinterpreting him, it seemed better in the first instance 

 to point out a well-known case in which precisely similar reason- 

 ing would lead to Lagrange's equations of motion where they are 

 known to be untrue (the reason, and a proper remedy, being 

 also generally known). This I did in your number of February 

 19 ; his reply, in the same number, is to the effect that he does 

 not intend to uphold the truth of Lagrange's equations in such 

 a case. It is not, however, logically permissible for anyone to 

 escape the inconvenient consequences of his own argument in 

 such a fashion. 



Possibly Mr. Heaviside has not grasped my point. If the 

 argument he puts forward on p. 298 is valid, I am unable to see 

 any point at which the following can without inconsistency be 

 alleged to fail : — " In the case of a rigid body rotating round a 

 fixed point with angular velocities wj, u„, co 3 about its principal 

 axes the kinetic energy T is a homogeneous quadratic function 

 of the to's, with coefficients which are constants. This makes 



2T = 



aT 



dT 



itw, au-.-, 



aT 



'3 



da. 



therefore 



• d (dT\ 



2 T = ON — I 1 + Of- 



1 dt\du l ) 

 But also by the structure of T, 



aT 



+ . 



dT dT dT 



,-1 1- Wo"; h u>» 



Vo,, 



dai % 



(S) 



(9) 



(10) 



d/d_T 

 3 d/\da s 



(") 



So, by subtraction of (10) from (9) 



• _ a(dT\ d/dT 

 ~ ai dt\d^J + U -dt\da, 2 



and therefore, by Newton, the torque about the first axis is the 

 coefficient of w, i.e. Ku v and similarly for the rest." 



There is no step in his demonstration which requires that the 

 coordinates should be " proper Lagrangian coordinates within 

 the meaning of the Act " ; in the proof usually given there is 

 such a step. 



It is with great diffidence, lest I may do Mr. Heaviside 

 injustice through misinterpreting him, that I now venture to 

 express the conjecture that in his argument he may possibly 

 have failed, as is sometimes done [by Maxwell, for instance, 

 " Treatise," second edition, § 561, equations (5)], to distinguish 

 between the displacements which a material system actually 

 receives during its motion and displacements which are perfectly 

 arbitrary subject only to the geometrical connections of the 

 system, and have thus confounded the equation 



_fd_ dT 



~\dl dv, 



X>, + 



dT\ 



which expresses that the rate at which work is done by the 

 forcives is equal to the rate at which the system gains kinetic 

 energy, with the very different one 



XjS*! + 



d_ 

 .dt' 



aT 

 dv. 



dT 

 dx. 



in which ix it &c, are arbitrary displacements as above. When 

 the latter equation is established, Lagrange's equations follow at 

 once, but Mr. Heaviside has made out no case for deducing 

 them from the former. In every case, as in the example I 

 cited, the right-hand member of the former equation can be 

 written in the form 



T/jIpjt-Vj, I',, »i, v„, v 2 , z/ 2 , 



• ) + 



in an infinite variety of ways, and accordingly it is sufficiently 

 obvious that there is no warrant for stating that the force on *, 

 is the coefficient of i', in any one such form more than in any 

 other. Samples of expressions which might thus be wrongly 

 obtained for the torque about the first axis in the instance 

 alluded to are 



A(i;, All, - (B- CJtOoWj, 



A<I>, + (B - C)w 2 ^3. Al "i - (BtD„ b - Ca^ 3 )/^. 



For the simpler case of a particle moving in a plane, one could 

 thus obtain, for example, the equations, 



X = m(x-if), \' = m{y + kx), 



where i is any quantity whatever. 



In short, the latter of the two equations compared above 

 differs from the former in being equivalent to a set of indepen- 

 dent equations equal in number to that of the coordinates of the 

 system. 



Similar remarks apply, of course, to his treatment of the 

 question of an elastic medium, p. 297. 



That the Principal of Energy, or of Activity, does not by 

 itself afford a sufficient basis from which to formulate the funda- 

 mental equations of dynamics in any form whatever is admitted 

 almost universally ; from Mr. Heaviside's letters it appears at 

 least doubtful whether he is willing to agree with this general 

 and well grounded opinion ; he has advanced no valid argument 

 against it, however. W. McF. Orr. 



February 22. 



A few weeks ago you published in a letter from Mr. 

 Heaviside a proof of Lagrange's equations of motion of a 

 system of bodies. I must confess that I in common with 

 others swallowed it, but I have now come to the conclusion, 

 that the proof, though doubtless admirable as an example of 

 the power of the " Principle of Activity," does not prove La- 

 grange's equations. In fact, if q be a coordinate, q the 

 corresponding velocity, and Q the corresponding force, we 

 have the result 



_ . f d ST 3T 1 



for any possible motion of the system. But we are not 

 entitled to equate the quantities in the brackets to zero, for 

 these are not independent 0/ q. The " proof " is, in fact, 

 merely Maxwell's well-known but fallacious proof, simpli- 

 fied by going direct instead of vi& Hamilton. 



Cambridge, February 28, R- F. W. 



Genius and the Struggle for Existence. 



Permit me to point out that Dr. A. R. Wallace's state- 

 ment (p. 296), " the comparatively short lives of million- 

 aire," is not supported by facts, at any rate by those for the 

 last three vears. 



The following has been obtained from the details con- 

 cerning estates on which death duties were paid. Nine 

 millionaires died during 1900, leaving in the aggregate 19 

 millions. The average age of these nine testators is seventy- 

 four — the youngest was fifty-nine and the oldest ninety-one 

 years. 



During 1901, we find that the deaths of eight millionaires 

 are recorded, whose joint estates were valued at io| millions. 

 In this case too, we find that the average age is above the 

 allotted threescore years and ten, being seventy-two. The 



NO. I740, VOL. 67] 



