January 29, 1903J 



NA TURE 



29/ 



herbarium were saved, passing into the hands of Nicola Cirillo 

 (1671-1734), a physician and botanist who possessed a private 

 botanical garden and was a Fellow of the Royal Society of 

 London, for which Society he collected data on the climate of 

 Naples, and wrote a treatise on the application of cold in the 

 treatment of fevers. Remaining in the Cirillo family, the 

 herbarium was finally bequeathed to the celebrated botanist 

 Domenico Cirillo, who preserved these volumes as the most 

 precious treasure in his collections. In 1783, Martin Vahl, a 

 friend of Linnisus, saw Imperato's herbarium in Cirillo's house, 

 and it is said that he fell on his knees in reverence before the 

 ancient relic. In 1799, when the royalist mob sacked Cirillo's 

 house and Cirillo himself was hanged, all his collections were 

 dispersed, including the herbarium of Imperato. Of the nine 

 volumes only one was saved, and finally came into the hands 

 of Camillo Minieri-Riccio, who in 1S63 published a short 

 account of this botanical relic (C. Minieri Riccio : " Breve 

 notizia dell' Erbario di Ferrante Imperato," Rendiconti delC 

 Accademia Pontaniana, xi., 1S63). Minieri says that Imperato's 

 name is written in the volume. 



The collections of Minieri-Riccio were finally sold to the 

 National Library at Naples, where the volume of Imperato's 

 herbarium may now be seen. 



The volume, of 268 pages, is bound in parchment and 

 is labelled " Collectio Plantarum Naturalium." It contains 440 

 plants, glued to the paper, each wilh one or more names. 

 There is an alphabetical index, probably written by Imperato 

 himself. 



The authorities in the Naples library do not seem aware of the 

 importance of the relic they possess, for the herbarium is kept 

 as an ordinary book and the plants are exposed to inevitable 

 damage and decay. Several of the specimens have already 

 been eaten up by insects. Italo Giglioli. 



R. Stazione Agraria Sperimentale, Rome, January 8. 



A Curious Projectile Force. 



I AM able to corroborate B.A. Oxon.'s letter (p. 247). In my 

 case, the screw stopper of the bottle (inverted) rested at an angle 

 against some books on a table. When the pressure of the gas 

 was sufficient to force out the stopper, the bottle sprang three 

 or four feet into the air and fell some distance off on the floor 

 of the 100m. Norman Lockyer. 



The Principle of Least Action. Lagrange's Equations. 



Whether good mathematicians, when they die, go to Cam- 

 bridge, I do not know. But it is well known that a large 

 number of men go there when they are young for the purpose 

 of being converted into senior wranglers and Smith's prizemen. 

 N iw at Cambridge, or somewhere else, there is a golden or 

 brazen idol called the Principle of Least Action. Its exact 

 locality is kept secret, but numerous copies have been made and 

 distributed amongst the mathematical tutors and lecturers at 

 Cambridge, who make the young men fall down and worship 

 the idol. 



I have nothing to say against the Principle. But I think a 

 great deal may be said against the practice of the Principle. 

 Truly, I have never practised it myself (except with pots and 

 pans), but I have had many opportunities of seeing how the 

 practice is done. It is usually employed by dynamicians to 

 investigate the properties of mediums transmitting waves, the 

 elastic solid for example, or generalisations or modifications of 

 the same. It is used to find equations of motion from energetic 

 data. I observe that this is done, not by investigating the 

 actual motion, but by investigating departures from it. Now 

 it is very unnatural to vary the time integral of the excess of 

 the total kinetic over the total potential energy to obtain the 

 equations of the real motion. Then again, it requires an in- 

 tegration over all space, and a transformation of the integral 

 beiore what is wanted is reached. This, too, is very unnatural 

 (though defensible if it were labour-saving), for the equation of 

 motion at a given place in an elastic medium depends only 

 upon its structure there, and is quite independent of the rest of 

 the medium, which may be varied anyhow. Lastly, I observe 

 that the process is complicated and obscure, so much so as to 

 easily lead to error. 



Why, then, is the P. of L. A. employed ? Is not Newton's 

 dynamics good enough ? Or do not the Leasl-Actionists know 

 that Newton's dynamics, viz. his admirable Fotce = Counter- 



NO. 1735, VOL. 67] 



force and the connected Activity Principle, can be directly 

 applied to construct the equations of motion in such cases as 

 above referred to, without any of the hocus pocus of departing 

 from the real motion, or the time integration, or integration 

 over all space, and with avoidance of much of the complicated 

 work. It would seem not, for the claim is made for the P. of 

 L. A. that it is a commanding general process, whereas the 

 principle of energy is insufficient to determine the motion. This 

 is wrong. But the P. of L. A. may perhaps be particularly 

 suitable in special cases. It is against its misuse that I write. 



Practical ways of working will naturally depend upon the 

 data given. We may, for example, build up an equation of 

 motion by hard thinking about the structure. This way is 

 followed by Kelvin, and is good, if the data are sufficient and 

 not too complicated. Or we may, in an elastic medium, assume 

 a general form for the stress and investigate its special properties. 

 Of course, the force is derivable from the stress. But the data 

 of the Least-Actionists are expressions for the kinetic and 

 potential energy, and the P. of L. A. is applied to them. 



But the Principle of Activity, as understood by Newton, 

 furnishes the answer on the spot. To illustrate this simply, let 

 it be only small motions of a medium like Green's or the same 

 generalised that are in question. Then the equation of 

 activity is 



div. qP = U + T; (1) 



that is, the rate of increase of the stored energy is the conver- 

 gence of the flux of energy, which is - qP, if q is the velocity 

 and P the stress operator, such that 



Pi = P 1 = iP n +jP 12 + kP 1:i (2) 



is the stress on the i plane. Here qP is the conjugate of Pq. 



By carrying out the divergence operation, (1) splits into two, 

 thus 



Fq = f, Gq = U. (3) 



Here F is a real vector, being the force, whilst G is a vector 

 force operator. Both have the same structure, viz. Pv, but in 

 F the differentiators in v act on P, whereas in G they are free 

 and act on q, if they act at all. 



Now when U is given, U becomes known. It contains q as 

 an operand. Knock it out ; then G is known ; and therefore F ; 

 and therefore the equation of motion is known, viz. 



F = - (wq), 

 at 



where m is the density, or the same generalised eolotropically, 

 or in various other ways which will be readily understood by 

 electricians who are acquainted with resistance operators. 



Of course, P becomes known also. So the form of U specifies 

 the stress, the translational force and the force operator of the 

 potential energy. To turn G to F is the same as turning 



. d , d\ 



A — to — 



dx dx 



If, for example, the displacement is D, the potential energy 

 is a quadratic function of the nine differentiants dD 1 /dx, &c. , of 

 the components. Calling these r u , r t „, &c. ; 



dU , ,.. dV 



dr. . 



u 



+ 



+ 



by the homogeneous property 

 idq/dy, 



r -, /rfU . d , dU . d , 

 V dr n ax (t>\ 



(4) 

 dr l2 



Therefore, since r 12 =ag 1 /dy= 



therefore, 



\d/- u dx dr u dy 

 writing P 21 for dU/dr v2 , 

 dP u + ,/P., 

 dy 



) 



q = Gq; 



\ dx dy dz ) 



dP-, + dP., + dP 3 



dx 



dy 



(5) 



(6) 



(7) 



It is clear that the differentiants in (4) (which involve the large 

 number 45 of coefficients of elasticity in the general case of 

 eolotropy) are the nine components of the conjugate of the 

 stress operator. Of course, vector analysis, dealing with the 

 natural vectors concerned, is the most suitable working agent, 

 but the same work may be done without it by taking the terms 

 involving q v q„, q 3 separately. 



Another expression for U is U = AGD, which shows how to 

 find F from U directly. 



