?68 



NATURE 



[February 19, 1903 



The Principle of Activity and Lagrange's Equations. 

 Rotation of a Rigid Body. 



There are some people who understand by Newton's second 

 >law of motion nothing more thin the three equations of motion 

 for a body which can be treated as a particle, viz., m 

 ■&.C. (or rather the equivalent equations for impulsive forces). 

 Such people, however, would probably not seriously object to 

 any dynamical truth whatsoever, from the conservation of 

 energy to the principle of varying action, being read into this 

 law, if only he who does so would explain clearly his own in- 

 terpretation of Newton's statement. I, for one, am a little 

 •curious to have stated fully the principle which justifies Mr. 

 Heaviside in his letter in your issue of January 29 in deducing 

 from the solitary equation 



Xdt ' dv~ dx^Y 1 ' ' ' 

 that " by Newton, the force on .v, is the coefficient of v v " 



It is a sufficient indication either of an incorrect premiss or 

 of bad logic, however obscure an argument may be, if the con- 

 clusion be wrong ; one does not readily see from Mr. Heaviside's 

 letter how he could object to his method being applied directly 

 to the motion of a rigid body with one point fixed, in which 

 case, as is well known, taking 



2T = AtOj C + Bo>._,- + Ca> s a 

 •it leads to a wrong expression for the external couple round the 

 axis of x, viz. Ai, instead of the correct one, Ac^ - ( B - C)<» a <u 3 . 



W. McF. Orr.' 

 Royal College of Science, Dublin, February 2. 



Prof. Orr's opening remarks perhaps indicate that the want 

 ■of appreciation of Newton's dynamics is even greater than I 

 supposed. My authority for Newton is that stiff but thorough- 

 going work, Thomson and Tait. On comparison, I find that 

 •Prof. Orr's "some people" seem to overlook the vitally im- 

 portant third law, without which there could be no dynamics 

 resembling the reality, and also the remarkable associated 

 scholium "Si ajstimatur . . .," enunciating the principle of 

 activity, which is of such universal and convenient application, 

 both by practicians and by some theorists. In my short outline 

 of the beginning of the theory of Lagrange's equations, my 

 argument " by Newton" referred to the activity principle. 



The example of failure given by Prof. Orr is remarkable 

 in more than one way. If the three coordinates specified the 

 configuration, then the equations of motion would come out in 

 the way indicated. It is clear, therefore, from the failure that 

 in the concrete example of a rotating rigid body, the coor- 

 dinates employed, which are the time-integrals of the angular 

 velocities about three moving axes, are not proper Lagrangian 

 coordinates within the meaning of the Act. If we use coor- 

 dinates which do fix the configuration (Thomson and Tait, 

 § 319), there is no failure. 



But it is quite easy to avoid the usual complicated trigono- 

 metrical work, and obtain the proper equations of motion by 

 allowing for the motion of the axes. Thus, if a is the angular 

 velocity, the angular momentum is 



rfT 



. . . = Aa^ + Ba.J + C^k, 



da-, 



-and the torque is its time differenliant, that is, 



dl 



du. 



F = Ai 1 i + Bfl.j + C<; a k + Aa 1 ' 7 i + P,„.." J r C' .."' 



dt ~ dt at 



' Here i, j, k are unit vectors specifying the directions of the 

 principal axes. They only vary by the rotation, sodi/d/ - Vai, 

 ■&C, and this makes 



r=Aa 1 (ja ;i -k(Z.,) + Ba,(k« l -ia,) + Ca 3 (i<r„-ja,)-rAa 1 i+ . . . 

 = f{te 1 -a a a 3 [B-C)}+j{. . . J + k{. . . }. 



This exhibits Euler's three well-known equations of motion 

 round the three principal moving axes. 



In general, T = ; \aMa, where M is a vectorial matrix (or 

 linear vector operator), fixed in the body. Then the momentum 

 is Ma, and the torque is 



F = Ma + Ma = Ma + ( VaM )a. 

 This allows M to be specified with respect to any axes fixed 

 in the rotating body. Of course, the principal axes are the 

 best. I may refer to my " Elec. Pa.," vol. ii., p. 547, footnote, for 



NO. I/38, VOL. 67] 



details of a similar calculation relating to the torque (and activity 

 thereof) produced in an eolotropic dielectric under electric stress. 

 The following concisely exhibits the necessity of allowing for 

 variation of M, and how it is done in the general case of » 

 independent variables: — Let T = ivMv = Apv. Then v is a 

 "vector" or complex of n velocities, and p = Mv is the 

 corresponding momentum, whilst M is a symmetrical matrix. 

 By differentiation to t, 



T = v(Mv+ \Mv) = Fv (Hamilton), 

 or 



T = v(p- aiv) = FT (Lagrange). 



Here F is the force on the system, in the same sense as v is 

 the velocity of the system. For M substitute v(dM/dx), to 

 come to the usual forms by breaking up into 11 components. 

 But the above are more general, because M may vary inde- 

 pendently of x. Activity should be the leading idea. 



Oliver Heaviside. 



Insects and Petal-less Flowers. 



I was much interested by Mr. Bulman's account of Prof. 

 Plateau's experiments in the matter of insects' visits to petal- 

 less flowers in the issue of Nature for February 5 (p. 3 '9), 

 wherein it is stated that Prof. Plateau contends that insects 

 " are not attracted by the brilliant colours of the blossoms, 

 but rather by the perception in some other way — probably 

 by scent— that there is honey or pollen." 



It has not been my good fortune to read Prof. Plateau's 

 own account of the experiments which led him to the above 

 conclusion, but it certainly appears to me, from your corre- 

 spondent's summary, that he is generalising from an observ- 

 ation which has only a strictly limited application. 



We are told that in the case of thirty poppies artificially 

 deprived of their petals, as compared with seventy intact 

 poppies, the average visits were as 45 is to 2'4, the most 

 sti iking case instanced being that of the Dipterous insect 

 Melanostoma mellina, the visits of which were as 4 is to o. 



The experiment and its result does not, to my mind, in the 

 least tend to bear out the theory it is advanced to support, 

 though your correspondent gives the method his approval. 



I do not wish to doubt the possibility of smell playing a 

 part in attracting insects, but I certainly cannot see that 

 the artificial removal of the coloured petals proves that colour 

 has no influence. We are fond of attributing great intelli- 

 gence and po^ver of perception to the bee, and yet in this 

 case the insect is not even given credit for being able to re- 

 cognise what are known to it, from possibly long experience, 

 as the essentia! parts of the flower ! Because we buy well 

 advertised goods, and still continue to buy them when their 

 proved virtue renders advertisement a thing of the past, is 

 it proof that the advertisement played no part in determin- 

 ing our choice? The answer is obvious. 



The greater number of insects visiting the poppies shorn 

 of their petals might easily be accounted for, especially in 

 the case of the Diptera, by the presence of some attractive 

 substance in the sap exuded from the cut tissues, and prob- 

 ably by the resulting greater accessibility. 



As a contrast to this experiment I would mention that 

 of Lord Avebury, which loses none of its significance through 

 being described in a popular magazine (the London, Christ- 

 mas number). Quantities of honey were taken and laid on 

 glass slips, and a marked bee was trained to come to a 

 certain spot for it. The honey was supplied on slips of six 

 different colours — blue, red, yellow, orange, green and white 

 — and on one plain slip. Lord Avebury so arranged matters 

 that the bee was persuaded to visit each and every slip 

 before returning to the hive, the method being as follows : — 



Seven slips in a row on lawn ; the bee arrives and alights 

 on (say) blue ; it is allowed to remain for a few seconds and 

 then driven off, the blue slip being withdrawn ; it then goes 

 to (say) white ; after a few seconds at white the bee is again 

 driven off, and goes to (say) yellow, the white slip being 

 also withdrawn ; after having visited all the slips in this 

 way, and being at last deprived of every one, the bee goes 

 back to the hive. 



During the bee's absence the glasses are replaced, but in 

 different order, and on the insect's return it is again noted 

 which slip receives first attention. 



Out of a hundred such complete rounds Lord Avebury 



