74 ADDRESS TO THE BRITISH ASSOCIATION, 1881. 



known as invariants, covariants, canonical forms, etc. By means of 

 these, mathematicians have arrived not only at many properties of the 

 quantics themselves, but also, at their application to physical pro- 

 blems. It would be a long and perhaps invidious task to enumerate 

 the many workers in this fertile field of research, especially in the 

 schools of Germany and of Italy ; but it is perhaps the less necessary 

 to do so, because Sylvester, aided by a young and vigorous staff at 

 Baltimore, is welding many of these results into a homogeneous mass 

 in the classical memoirs which are appearing from time to time in the 

 American ' Journal of Mathematics.' 



In order to remove any impression that these extensions of algebra 

 are merely barren speculations of ingenious intellects, I may add 

 that many of these derivative forms, at least in their elementary 

 stages, have already found their way into the text-books of mathe- 

 matics ; and one class in particular, known by the name of deter- 

 minants, is now introduced as a recognised method of algebra, greatly 

 to the convenience of all those who become master of its use. 



In the extension of mathematics it has happened more than once 

 that laws have been established so simple in form, and so obvious in 

 their necessity, as scarcely to require proof. And yet their applica- 

 tion is often of the highest importance in checking conclusions which 

 have been drawn from other considerations, as well as in leading to 

 conclusions which, without their aid, might have been difficult of at- 

 tainment. The same thing has occurred also in physics ; and notably 

 in the recognition of what has been termed the ' Law of the Conserva- 

 tion of Energy.' 



Energy has been defined to be ' The capacity, or power, of any 

 body, or system of bodies, when in a given condition, to do a measur- 

 able quantity of work.' Such work may either change the condition 

 of the bodies in question, or it may affect other bodies ; but in either 

 case energy is expended by the agent upon the recipient in perform- 

 ance of the work. The law then states that the total amount of 

 energy in the agents and recipients taken together remains unaltered 

 by the changes in question. 



Now the principle on which the law depends is this : ' that every 

 kind of change among the bodies may be expressed numerically in one 

 standard unit of change, viz., work done, in such wise that the result 

 of the passage of any system from one condition to another may be 

 calculated by mere additions and substractions, even when we do not 

 know how the change came about. This being so, all work done by 

 a system may be expressed as a diminution of energy of that system, 



