July 3. 1879] 



NATURE 



215 



of happiness or misery, knowledge or ignorance, and the 

 dynamical transactions between them may or may not be 

 accompanied with the conscious effort which the word 

 force suggests to us when we imagine one of the bodies to 

 be our own, but so long as their motions are related to each 

 other according to the conditions laid down in dynamics, 

 we call them, in a perfectly intelligible sense, dynamical 

 or material systems. 



In this, the second edition, we notice a large amount of 

 new matter, the importance of which is such that any 

 opinion which we could form within the time at our 

 disposal would be utterly inadequate. But there is one 

 point of vital importance in which we observe a marked 

 improvement, namely, in the treatment of the generalised 

 equations of motion. 



Whatever may be our opinion about the relation of 

 mass, as defined in dynamics, to the matter which consti- 

 tutes real bodies, the practical interest of the science 

 arises from the fact that real bodies do behave in a 

 manner strikingly analogous to that in which we have 

 proved that the mass-systems of abstract dynamics must 

 behave. 



In cases like that of the planets, when the motions we 

 have to account for can be actually observed, the equa- 

 tions of Maclaurin, which are simply a translation of 

 Newton's laws into the Cartesian system of co-ordinates, 

 are amply sufficient for otir purpose. But when we have 

 reason to believe that the phenomena which fall under 

 our observation form but a very small part of what is 

 really going on in the system, the question is not — what 

 phenomena wiU result from the hypothesis that the 

 system is of a certain specified kind? but — what is the most 

 general specification of a material system consistent with 

 the condition that the motions of those parts of the 

 system which we can observe are what we find them 

 to be? 



It is to Lagrange, in the first place, that we owe the 

 method which enables us to answer this question without 

 asserting either more or less than all that can be legiti- 

 mately deduced from the observed facts. But though 

 this method has been in the hands of mathematicians 

 since 1788, when the M^caniqite Analytiqtte was pub- 

 lished, and though a few great mathematicians, such as 

 Sir W. R. Hamilton, Jacobi, &c., have made important 

 contributions to the general theory of dynamics, it is 

 remarkable how slow natural philosophers at large have 

 been to make use of these methods. 



Now, however, we have only to open any memoir on a 

 physical subject in order to see that these dynamical 

 theorems have been dragged out of the sanctuary of pro- 

 found mathematics in which they lay so long enshrined, 

 and have been set to do all kinds of work, easy as well as 

 difficult, throughout the whole range of physical science. 



The credit of breaking up the monopoly of the great 

 masters of the spell, and making all their charms familiar 

 in our cars as household words, belongs in great measure 

 to Thomson and Tait. The two northern wizards were 

 the first who, without compunction or dread, uttered in 

 their mother tongue the true and proper names of those 

 dynamical concepts which the magicians of old were wont 

 to invoke only by the aid of muttered symbols and inar- 

 ticulate equations. And now the feeblest among us can 

 repeat the words of power and take part in dynamical 



discussions which but a few years ago we should have left 

 for our betters. 



In the present edition we have for the first time an ex- 

 position of the general theory of a very potent form of 

 incantation, called by our authors the Ignoration of Co- 

 ordinates. We must remember that the co-ordinates of 

 Thomson and Tait are not the mere scaffolding erected 

 over space by Descartes, but the variables which deter- 

 mine the whole motion. We may picture them as so 

 many independent driving-wheels of a machine which has 

 as many degrees of freedom. In the cases to which 

 the method of ignoration is applied there are certain 

 variables of the system such that neither the kinetic 

 nor the potential energy of the system depends on 

 the values of these variables, though of course the 

 kinetic energy depends on their momenta and velo- 

 cities. The motion of the rest of the system cannot 

 in any way depend on the particular values of these 

 variables, and therefore the particular values of these 

 variables cannot be ascertained by means of .'.ny observa- 

 tion of the motion of the rest of the system. We have 

 therefore no right, from such observations, to assign to 

 them any particular values, and the only scientific way of 

 dealing with them is to ignore them. 



But this is not all. Since these variables do not appear 

 in the expression for the potential energy, there can be 

 no force acting on them, and therefore their momenta are, 

 each of them, constant, and their velocities are functions 

 of the variables, but, since their own variables do not 

 enter into the expressions, we may consider them as 

 functions of the other variables, or, as they are here 

 called, the retained co-ordinates, and of the constant 

 momenta of the ignored co-ordinates. 



From the velocities as thus expressed, together with 

 the constant momenta, we obtain the contribution of the 

 ignored co-ordinates to the kinetic energy of the system 

 in terms of the retained co-ordinates and of the constant 

 momenta of the ignored co-ordinates. This part of the 

 kinetic energy, being independent of the velocities of the 

 retained co-ordinates, is, as regards the retained co- 

 ordinates, strictly positional^ and may be considered for 

 all experimental purposes as if it were a term of the 

 potential energy. The other part of the kinetic energy is 

 a homogeneous quadratic function of the velocities of the 

 retained co-ordinates. In the final equations of motion 

 neither the ignored co-ordinates nor their velocities ap- 

 pear, but everything is expressed in terms of the retained 

 co-ordinates and their velocities, the coefficients, however, 

 being, in general, functions of the constant momenta of 

 the ignored co-ordinates. 



We may regard this investigation as a mathematical 

 illustration of the scientific principle that in the study of 

 any complex object, we must fix our attention on those 

 elements of it which we are able to observe and to cause 

 to vary, and ignore those which we can neither observe 

 nor cause to vary. 



In an ordinary belfry, each bell has one rope which 

 comes down through a hole in the floor to the bellringers' 

 room. But suppose that each rope, instead of acting on 

 one bell, contributes to the motion of many pieces of 

 machinery, and that the motion of each piece is deter- 



I The division of forces into motional and positional is introduced at 

 p. 370. 



