442 THEORY OF HEAT. [CHAP. IX. 



general theory of equations, and requires only that we should 

 suppose known the form of the imaginary roots which every equa 

 tion may have. We have not referred to it in this work, since its 

 place is supplied by constructions which make the proposition more 

 evident. Moreover, we have treated a similar problem analytically, 

 in determining the varied movement of heat in a cylindrical body 

 (Art. 308). This arranged, the problem consists in discovering 

 numerical values for a lt # 2 , a g ,...a f , &c., such that the second 

 member of equation (e) necessarily becomes equal to xF(x), when 

 we substitute in it for x any value included between and the 

 whole length X. 



To find the coefficient ., we have multiplied equation (e) by 

 dx sin fi t a;, and then integrated between the limits x 0, x = X, 

 and we have proved (Art. 291) that the integral 



rX 



I dx sin figc sin ^x 



Jo 



has a null value whenever the indices i and j are not the same; 

 that is to say when the numbers p i and /*, are two different roots 

 of the equation (/). It follows from this, that the definite inte 

 gration making all the terms of the second member disappear, 

 except that which contains a it we have to determine this coefficient, 

 the equation 



x ix 



dx \x F (x\ sin pp] = a.l dx sin pp sin pp. 



o Jo 



Substituting this value of the coefficient a t in equation (e), we 

 derive from it the identical equation (e), 



x 

 dot. a,F(a) s 



r 

 I 



Jo 



r 

 I 



d@ sin a & sin a B 



Jo 



aux equations transcendantes , by Fourier. The author shews that the imaginary 

 roots of sec x=Q do not satisfy the equation tance=0, since for them, tan# = JN / - 1. 

 The equation tan x = is satisfied only by the roots of sin x 0, which are all real. 

 It may be shewn also that the imaginary roots of sec # = do not satisfy the equation 

 x-mtsinx-Q, where m is less than 1, but this equation is satisfied only by the 

 roots of the equation f(x) = x cos x - m s mx = 0, which are all real. For if 

 fr +1 (x), f r (x], f r -i(x), are three successive differential coefficients of f(x), the values 

 of x which make f r ()=0, make the signs of / r+1 (x) and / r-1 (x) different. Hence 

 by Fourier s Theorem relative to the number of changes of sign of f(x) and its 

 successive derivatives, /(.r) can have no imaginary roots. [A. F.j 



