430 THEORY OF HEAT. [CHAP. IX. 



factory under the symbol sin/&amp;gt; (OL X). These considerations are 

 too well known to make it necessary to recall them. 



Above all, it must be remarked that the function /(a?), to which 

 this proof applies, is entirely arbitrary, and not subject to a con 

 tinuous law. We might therefore imagine that the enquiry is 

 concerning a function such that the ordinate which represents it 

 has no existing value except when the abscissa is included between 

 two given limits a and b, all the other ordinates being supposed 

 nothing ; so that the curve has no form or trace except above the 

 interval from x = a to x = b, and coincides with the axis of a in 

 all other parts of its course. 



The same proof shews that we are not considering here infinite 

 values of x, but definite actual values. We might also examine on 

 the same principles the cases in which the function f(x) becomes 

 infinite, for singular values of x included between the given limits; 

 but these have no relation to the chief object which we have in 

 view, which is to introduce into the integrals arbitrary functions ; 

 it is impossible that any problem in nature should lead to the 

 supposition that the function f(x) becomes infinite, when we 

 give to a; a singular value included between given limits. 



In general the function f(x) represents a succession of values 

 or ordinates each of which is arbitrary. An infinity of values being 

 given to the abscissa x, there are an equal number of ordinates 

 / (x). All have actual numerical values, either positive or negative 

 or nul. 



We do not suppose these ordinates to be subject to a common 

 law; they succeed each other in any manner whatever, and each of 

 them is given as if it were a single quantity. 



It may follow from the very nature of the problem, and from 

 the analysis which is applicable to it, that the passage from one 

 ordinate to the following is effected in a continuous manner. But 

 special conditions are then concerned, and the general equation (B), 

 considered by itself, is independent of these conditions. It is 

 rigorously applicable to discontinuous functions. 



Suppose now that the function f(x) coincides with a certain 

 analytical expression, such as sina, e~ x \ or $ (x), when we give to 

 x a value included between the two limits a and b, and that all 



