Definite Integrals, with Physical Applications. Sbb 



function under the definite integral, when the former is a rational 

 function, we have only to multiply it by t" and taking the 



coefficient of- in the product, divide it by t. This theorem, proved 



in Section (1), is there also extended to any number of variables; 

 the remaining part of the Section illustrates the nature of the 

 application of the theorem, and includes the mode of treating 

 irrational functions. 



The functions considered in the first Section are all continuous, 

 the mode of treating discontinuous functions is shewn in the 

 second ; it was necessary to attend to this class, because the 

 phenomena presented by nature are mostly of that kind. Thus 

 the action of developed electricity follows two different laws, 

 according as the point acted on, is within or without the surface 

 of the body ; the function which expresses this action generally, 

 must therefore be discontinuous, the interruption taking place 

 at the surface. In like manner heat is propagated simultaneously, 

 in the earth, the sea, and the atmosphere, but according to dif- 

 ferent laws in each, and other instances of discontinuity are 

 observable in every department of physics. 



Now we shall obviate the difficulty thus presented by discon- 

 tinuous functions, if we can obtain a formula, which without 

 any alteration of form may continue to represent the function, 

 under all circumstances. This object is attained by a simple 

 application of the theory of Algebraic Equations. For when an 

 Equation is resolved by the method given in my former paper*, 

 the expression for the least root is in a form symmetrical with 

 respect to all the roots. If for simplicity we only consider two 



• Cambridge Philosophical Transactions, Vol. IV. 

 Vol. IV. Part III. Z z 



