SECT. IV.] SYSTEM OF QUANTITIES REPRESENTED. 447 



426. We see clearly by this examination that the function /(.r) 

 represents, in our analysis, the system of a number n of separate 

 quantities, corresponding to n values of x included between and 

 X, and that these n quantities have values actual, and consequently 

 not infinite, chosen at will. All might be nothing, except one, 

 whose value would be given. 



It might happen that the series of the n values f lt f 2 ,f s .../ 

 was expressed by a function subject to a continuous law. such as 

 x or x 3 , sin#, or cos a-, or in general &amp;lt;j&amp;gt; (x) ; the curve line 0(70, 

 whose ordinates represent the values corresponding to the abscissa 

 x, and which is situated above the interval from x = to x = X, 

 coincides then in this interval with the curve whose ordinate is 

 &amp;lt;/&amp;gt; (x), and the coefficients a lt a 8 , a 3 ... a n of equation (e) determined 

 by the preceding rule always satisfy the condition, that any value 

 of x included between and X, gives the same result when substi 

 tuted in &amp;lt;p (x)-, and in the second member of equation (e). 



F(x) represents the initial temperature of the spherical shell 

 whose radius is x. &quot;We might suppose, for example, F(x) = bx, 

 that is to say, that the initial heat increases proportionally to the 

 distance, from the centre, where it is nothing, to the surface 

 where it is bX. In this case xF(x) or f(x) is equal to bx 2 ; and 

 applying to this function the rule which determines the coeffi 

 cients, bx* would be developed in a series of terms, such as 



a l sin fax) + a 2 sin fax) + a z sin fax) + ... + a n sin fax). 



Now each term sinQ^oj), when developed according to powers 

 of x, contains only powers of odd order, and the function bx* is 

 a power of even order. It is very remarkable that this function 

 bx z , denoting a series of values given for the interval from 

 to X, can be developed in a series of terms, such as a t sin fax). 



We have already proved the rigorous exactness of these 

 results, which had not yet been presented in analysis, and we 

 have shewn the true meaning of the propositions which express 

 them. We have seen, for example, in Article 223, that the 

 function cos# is developed in a series of sines of multiple arcs, 

 so that in the equation which gives this development, the first 

 member contains only even powers of the variable, and the second 

 contains only odd powers. Reciprocally, the function sin x, into 



