SECT. II.] EQUATION OF CONDITION HAS ONLY REAL ROOTS. 289 



The exact value of the ratio has been already determined 1 . 

 The problem of the movement of heat in a sphere includes that 

 of the terrestrial temperatures. In order to treat of this problem 

 at greater length, we have made it the object of a separate 

 chapter 8 . 



305. The use which has been made above of the equation 

 = X is founded on a geometrical construction which is very 



well adapted to explain the nature of these equations. The con 

 struction indeed shows clearly that all the roots are real ; at the 

 same time it ascertains their limits, and indicates methods for 

 determining the numerical value of each root. The analytical 

 investigation of equations of this kind would give the same results. 

 First, we might ascertain that the equation e X tan e = 0, in 

 which X is a known number less than unity, has no imaginary 

 root of the form m + njl. It is sufficient to substitute this 

 quantity for e ; and we see after the transformations that the first 

 member cannot vanish when we give to m and n real values, 

 unless n is nothing. It may be proved moreover that there can 

 be no imaginary root of any form whatever in the equation 



A e cos X sin e 

 e X tan e = 0. or = 0. 



cose 



In fact, 1st, the imaginary roots of the factor = do not 



cose 



belong to the equation e X tan e = 0, since these roots are all of 



the form m + nj 1 ; 2nd, the equation sin e - cos e = has 



X 



necessarily all its roots real when X is less than unity. To prove 

 this proposition we must consider sin e as the product of the 

 infinite number of factors 



1 It is 9 : &=i*X* : e^Y 2 , as may be inferred from the exponent of the first 

 term in the expression for z, Art. 301. [A. F.] 



2 The chapter referred to is not in this work. It forms part of the Suite du 

 inemorie sur la theorie du mouvement de la chaleur dans les corps solides. See note, 

 page 10. 



The first memoir, entitled Theorie du mouvf.ment de la chaleur dans les corps 

 solides, is that which formed the basis of the Theorie analytique du mouvement de 

 la chaleur published in 1822, but was considerably altered and enlarged in that 

 work now translated. [A. F.] 



F. H. 19 



