SECT. IV.] SEPARATE FUNCTIONS. 455 



certain form of the integral which was denoted as general ; and 

 equation (7) of Art. 398 was propounded under this designa 

 tion ; but this distinction has no foundation, and the use of a 

 single integral would only have the effect, in most cases, of com 

 plicating the investigation unnecessarily. It is moreover evident 

 that this integral (7) is derivable from that which we gave in 1807 

 to determine the movement of heat in a ring of definite radius E ; 

 it is sufficient to give to R an infinite value. 



llth. It has been supposed that the method which consists in 

 expressing the integral by a succession of exponential terms, and 

 in determining their coefficients by means of the initial state, 

 does not solve the problem of a prism which loses heat unequally 

 at its two ends ; or that, at least, it would be very difficult to 

 verify in this manner the solution derivable from the integral (7) 

 by long calculations. We shall perceive, by a new examination, 

 that our method applies directly to this problem, and that a single 

 integration even is sufficient 1 . 



12th. We have developed in series of sines of multiple arcs 

 functions which appear to contain only even powers of the variable, 

 cos a; for example. We have expressed by convergent series or 

 by definite integrals separate parts of different functions, or func 

 tions discontinuous between certain limits, for example that which 

 measures the ordinate of a triangle. Our proofs leave no doubt 

 of the exact truth of these equations. 



13th. We find in the works of many geometers results and pro 

 cesses of calculation analogous to those which we have employed. 

 These are particular cases of a general method, which had not yet 

 been formed, and which it became necessary to establish in order 

 to ascertain even in the most simple problems the mathematical 

 laws of the distribution of heat. This theory required an analysis 

 appropriate to it, one principal element of which is the analytical 

 expression of separate functions, or of parts of functions. 



By a separate function, or part of a function, we understand a 

 function / (x) which has values existing when the variable x is 

 included between given limits, and whose value is always nothing, 

 if the variable is not included between those limits. This func 

 tion measures the ordinate of a line which includes a finite arc of 

 1 See the Memoir referred to in note 1, p. 12. [A. F.] 



