April 5, 1894] 



'NATURE 



529 



Nature Pictures for Little People. By W. Mawer, and 

 others. (London : The Sunday School Association. 



Some parts of this book are very good ; others impress 

 us much less favourably. In order to test whether the 

 little people for whom the book is intended were interested 

 in its contents, we gave it to a few average boys to read, 

 and found that their verdict was the same as that ex- 

 pressed above. The authors frequently assume that their 

 young readers are familiar with things not commonly 

 seen, and with expressions not usually found in children's 

 reading-books. For instance, one section of the book is 

 headed 'ATvrtpv^ and begins with the sentence, " Rara 

 avis in terris indeed ! " This is very well in its way, but 

 is out of place in a book of this character. Also, the 

 numerous small witticisms and puns do not add to the 

 interest or value of the descriptions. " You will wonder,'' 

 we find in an account of whales "notwithstanding all 

 you have read, how the big whale in the picture was 

 drawn out of the sea and placed upon that rock. I have 

 an opinion of my own upon the subject, and will confide 

 it to you : the artist i^/'£7a him up there." The following 

 specimen is also unattractive, if not misleading. "When 

 collecting those tiny shells which you find ready per- 

 forated for threading into necklaces, do you wonder 7f-^j/ 

 they are so perforated ? If so, let me tell you. The 

 little creature which lives in t"he shell is the favourite food 

 of those bigger ones which have been introduced to you 

 as Roaring Buckles — namely, whelks, as well as purples : 

 and to get out the sweet morsel — oh ! so sweet — with 

 their tongues (or what you may call their tongues) they 

 file out the little round hole, and then — oh ! " There is 

 much brilliant writing of this character, the style being 

 after that in Kingsley's " Water Babies," and a very long 

 way after. In our opinion, however, the best parts 

 are those not containing composition of the kind 

 quoted. Several excellent illustrations, and two or three 

 simply-worded and interesting sections, are the book's 

 only redeeming features. 



LETTERS TO THE EDLTOR. 



{The Editor does not hold himself responsible for opinions ex- 

 pressed by his correspondents. Neither can he undertake 

 to return, or to correspond with the writers of, rejected 

 manuscripts intetided for this or any other part c/ Nature. 

 No notice is taken of anonymous com)nunications.'\ 



The Foundations of Dynamics. 



Dynamical investigations are often made to depend upon 

 the following three propositions, viz., the principle of linear 

 tncmentn7>i, the principle of angular moineiiluin, and the prin- 

 ciple of energy. In treatises which are concerned with ordinary 

 mechanical systems, such as rigid bodies, elastic solids and 

 friitionless fluids, the word energy is generally employed in the 

 restricted sense of mechanical energy, and is confined to two 

 particular species, viz. kinetic energy arising from such motions 

 of the system as can be controlled by ordinary mechanical 

 agencies, and potential energy arising from the configuration 

 of the system or from its position in space. All other iorms of 

 energy, such as molecular kinetic energy arising from the pro- 

 duction of heat by friction, and chemical potential energy 

 contained in fuel, explosive compounds, &c., are excluded from 

 consideration. In systems of this kind the sum of the kinetic 

 and potential energies is an invariable quantity, and this pro- 

 position may be termed the principle of the conservation of 

 mechanical energy. 



It is important to recollect that the principle of momentum 

 is a proposition of a wider character than the principle of the 



NO. 1275. VOL. 49] 



conservation of mechanical energy, for the former proposition 

 is true when the system possesses viscosity or internal friction 

 which gives rise to a conversion of mechanical energy into heat. 

 From this fact it follows that the principles of "momentum and 

 energy cannot be regarded as axiomatic, but depend uix-n 

 certain propositions of a more fundamental character, which 

 ought to be capable of explaining why it is that the principle of 

 momentum is true in the case of viscous systems, whilst the 

 principle of the conservation of mechanical energy does not hold 

 good. 



In order to examine this question we shall start with New- 

 ton's three Laws of Motion, which will be regarded as funda- 

 mental axioms, and shall first inquire how far they will carry 

 us ; and we shall find that when we consider the motion of 

 masses of matter of finite size, Newton's Laws are not sufficient 

 to enable us to determine the motion unless the matter is in the 

 ideal state of a frictionless or perfect fluid. In all other cases 

 an additional hypothesis is necessary. 



To clear the ground, it may be well to point out that the 

 parallelogram of forces is a direct consequence of Newton's 

 second law, and that the parallelogram of couples is a conse- 

 quence of the parallelogram of forces ; whence all the pro- 

 positions relating to the composition of forces and couples, in- 

 cluding the one which enables any system of forces and couples 

 to be compounded into a wrench, are deductions from the 

 second law. 



There are two methods of investigating the motion of a mass 

 of matter of finite size. In the first place, we may suppose the 

 matter to consist of very small discrete masses under the influ- 

 ence of molecular forces, together v/ith bodily forces, such as 

 gravity and the like ; in the second place, we may regard the 

 mass as a continuous one which may be subdivided into small 

 differential elements of volume. We shall consider the subject 

 from the first point of view. 



Let ;«j, ;«2 be the masses of any two elements (xj, jj/j, c,), 

 (.Tj, J'.,, 2,), their co-ordinates referred to yfxt'i^axes; let Rj, 

 be the molecular force of 7n.^ on m-^, R^i of m-^ on Wj ! ^Iso let 

 Fj be the bodily force acting on m^. 



By Newton's second law, the forces Rj.,, F, may be resolved 

 into components f-^^, g^^i ^^ii^ ^^d Xj, Yj, Zj parallel to the 

 axes ; also by the same law, the equations of motion of the 

 elements are 



m^x^ = Xi -h/i,. -r/i3 + 



m^^^ X, -1-7,1 +y;3 + 



"hf'i = "^'l +i'l2 + giz^ 



••} 



(I) 



It follows from Newton's third law that the molecular force 

 of m., on 7«i is equal in magnitude and opposite in direction to 

 that of /«! on w., ; whence/j., = - fi, &c. Accordingly, if we 

 add the system of equations (i) it follows that all the molecular 

 forces disappear, and we obtain 



2(w.V) = 2(X) (3) 



Equation (3) is the analytical expression for the principle of 

 linear momentum, and from the above investigation it follows 

 that this principle is a direct consequence of the second and 

 third laws. 



Since the equations of motion of a perfect fluid can be deduced 

 from the above principle, it follows that the whole theory of 

 the hydrodynamics of frictionless fluids depends solely on New- 

 ton's second and third laws. 



We must next consider the principle of angular momentum. 

 From (i) and (2) we obtain 



'S.m{xy ~ y.v) = 2(xY - yX) + g^^x^ + g„_^x., - f^.y^ - f-^y^ + 

 (4) 



By the third law the last four terms may be written 



fii{y-2 - Ji) - i'i2(-^2 - -^l) (5) 



which represents one of the components of the couple due to 

 the mutual action of /n^ and ;«.,. 



Now Newton's third law states that the action of m^ on w._, 

 is equal and opposite to that of Wo on m^ ; but it makes no 

 assertion to the effect that the mutual action consists of a force 

 acting along the Hue joining than. An assumption of this kind 

 would limu the generality of the law, and would be one of a 

 somewhat doubtful character, for in viscous systems there are 

 grounds for thinking that this mutual action may consist in part 



