426 THEORY OF HEAT. [CHAP. IX. 



following proofs, which we gave in our first researches, are very 

 suitable to exhibit the truth of these propositions. 



In the general equation 



-i r+x&amp;gt; /+&amp;lt; 



f(x) = - I cfaf (a) dp cos (py. -px) t 



&quot; oo JO 



which is the same as equation (B), Art. 404, we may effect the in 

 tegration with respect to p, and we find 



a-x 



We ought then to give to p, in the last expression, an infinite 

 value; and, this being done, the second member will express the 

 value of f(&). We shall perceive the truth of this result by 

 means of the following construction. Examine first the definite 



C m vi / y* 



integral I dx - , which we know to be equal to JTT, Art. 356. 

 Jo x 



If we construct above the axis of x the curve whose ordinate is 

 sin x, and that whose ordinate is -, and then multiply the ordinate 



M&amp;gt; 



of the first curve by the corresponding ordinate of the second, we 

 may consider the product to be the ordinate of a third curve 

 whose form it is very easy to ascertain. 



Its first ordinate at the origin is 1, and the succeeding ordinates 

 become alternately positive or negative; the curve cuts the axis 

 at the points where x = TT, 2?r, 3?r, &c., and it approaches nearer 

 and nearer to this axis. 



A second branch of the curve, exactly like the first, is situated 



r sin x 



to the left of the axis of y. The integral I dx is the area 



Jo af 



included between the curve and the axis of x, and reckoned from 

 x up to a positive infinite value of x. 



00 



The definite integral / dx , in which p is supposed to be 

 Jo & 



any positive number, has the same value as the preceding. In 



fact, let px = z ; the proposed integral will become I dz , and, 



Jo z 

 consequently, it is also equal to ^TT. This proposition is true, 



