PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 645 



be employed only when it is necessary to start from first principles. 

 All mates which have once been determined may be taken from the 

 latter sections of the table with a great saving of time and energy. 

 Part 2 of the table shows the elementary transformations and com- 

 binations of pairs ; these theorems may be employed either to extend 

 a given table of coefficient pairs, or to cover a given group of coeffi- 

 cients with a shorter table of specific pairs. It is assumed that anyone 

 desiring to make serious use of the table will first become familiar 

 with these elementary combinations and transformations; even the 

 simple addition, factor and transposition theorems (201), (205), (217) 

 are most useful. 



Part 3 of the table contains seven pairs, which are called key pairs 

 because all specific pairs listed in the entire table may apparently be 

 derived from them by specialization or by passing to a limit after any 

 necessary use has been made of the elementary combinations and trans- 

 formations of Part 2, amplified, as indicated, by the removal of certain 

 unnecessary restrictions to real quantities. If an assigned coefficient 

 is not included in Part 3 as thus generalized, then this coefficient cannot 

 be found anywhere in Table I. Part 3, therefore, serves the useful 

 purpose of giving a bird's-eye view of the entire table. The seven 

 pairs are presumably redundant as they stand. 



For applicational purposes it is most desirable to have a table 

 which lists the precise pair required; many special cases which have 

 been used in practical applications may be found in Parts 4-9 of 

 Table I which constitute a short classified list of particular cases. 

 It is important to remember that a given coefficient should be looked 

 for on the other side of the table if it is not found on its own side since 

 all pairs are transposable by (217) or (218). In the tables as they 

 stand, some pairs have been transposed, but this is not true in the 

 majority of cases. 



Whenever an infinite process is to be employed, such as infinite 

 series, integration or differentiation, the permissibility of the process 

 is a question which must be answered for the particular case in hand ; 

 the formal result given in Table I may break down, for example, if 

 either the original or the transformed pair is a singular pair. This 

 general warning necessarily applies to every part of the subject of 

 coefficient pairs just because it is a part of the general subject of mathe- 

 matical analysis. 



It is intended that the statement of each pair in the entire table 

 shall eventually include every limitation and every warning which 

 the mathematical sponsors for that pair would consider necessary to 

 guarantee its safe use by anyone understanding the fundamental 



