SECT. III.] LIMITS OF THE VARIABLE. 153 



The number k which enters into these equations is not the 

 same for all, and it represents in each one a certain quantity 

 which is always included between 1 and 1 ; m is equal to the 

 number of terms of the series 



cos x - cos 3# + - cos 5x . . . -^ cos (2m 1) x t 

 o 5 ~ili 1 



whose sum is denoted by y. 



188. These equations could be employed if the number m 

 were given, and however great that number might be, we could 

 determine as exactly as we pleased the variable part of the value 

 of y. If the number m be infinite, as is supposed, we consider 

 the first equation only; and it is evident that the two terms 

 which follow the constant become smaller and smaller; so that 

 the exact value of 2y is in this case the constant c; this constant 

 is determined by assuming x = in the value of y, whence we 

 conclude 



-- = COS X = COS Sx + - COS DX ;= COS 7# + T: COS 9.E &C. 



4 3 o 7 9 



It is easy to see now that the result necessarily holds if the 

 arc x is less than \ir. In fact, attributing to this arc a definite 

 value X as near to JTT as we please, we can always give to in 



a value so great, that the term - (sec a; sec 0), which completes 



the series, becomes less than any quantity whatever ; but the 

 exactness of this conclusion is based on the fact that the term 

 sec x acquires no value which exceeds all possible limits, whence 

 it follows that the same reasoning cannot apply to the case in 

 which the arc x is not less than JTT. 



The same analysis could be applied to the series which express 

 the values of Ja?, log cos x, and by this means we can assign 

 the limits between which the variable must be included, in order 

 that the result of analysis may be free from all uncertainty ; 

 moreover, the same problems may be treated otherwise by a 

 method founded on other principles 1 . 



189. The expression of the law of fixed temperatures in 

 a solid plate supposed the knowledge of the equation 



1 Cf. De Morgan s Eiff. and Int. Calculus, pp. 605 609. [A. F.] 



