ADDEESS. 41 



classical memoirs wliicli arc appearing from time to time iu the 

 ' American Journal of Mathematics.' 



In order to I'emove any impression that these extensions of algebra 

 are mei-ely barren speculations of ingenious intellects, I may add that, 

 many of these derivative forme, at least in their elementary stages, have 

 already found their way into the text-books of mathematics ; and one 

 class in particular, known by the name of determinants, is now introduced 

 as a recognised method of algebra, greatly to the convenience of all those 

 who become masters 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 application is often 

 of the highest importance in checking conclusions which have been 

 di-awn fi'om other considerations, as well as in leading to conclusions 

 which, without their aid, might have been difficult of attainment. The 

 same thing has occurred also in physics ; and notably in the recognition 

 of what has been termed the ' Law of the Conservation 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 measurable 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 performance 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 subtractions, 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, and all work done 

 upon a system as an accession of energy. Consequently, the energy lost 

 by one system in performance of work will be gained by another in having 

 work done upon it, and the total energy, as between the two systems, will 

 remain unchanged. 



There are two cases, or conditions, of energy which, although sub- 

 stantially the same, are for convenience regarded separately. These may 

 be illustrated by the following example. Work may be done upon a body, 

 and energy communicated to it, by setting it in motion, e.g. by lifting it 

 against gravity. Suppose this to be done by a spring and detent ; and 

 suppose further the body, on reaching its highest point, to be caught so 

 as to rest at that level on a support. Then, whether we consider the body 

 at the moment of starting, or when resting on the support, it has equally 

 received an accession of energy from the spring, and is therefore equally 

 capable of communicating energy to a third body. But in the one case 

 this is due to the motion which it has acquired, and in the other to the 



