14:2 On Operators in Physical Mathematics. [June 1, 



Details concerning the above Relations. 



73. As regards [d], ifc may be obtained from [a], [6], or [c]. 

 These have not yet been done, so a little detail is now given. Thus, 

 from [6] to [d] : 



* r i + i>3 



F? + ""}(^ 

 _ 1 i__ 



In the first line we expand the function H ; to get the second 

 line we integrate with unit operand ; and, finally, let ei r operate to 

 get (175). 



74. Next, from [c] to [d] : 





which needs no explanation, as the course is similar to the previous 

 leading to (175). 



75. As regards deriving [d] from [a], this may be done by har- 

 monic decomposition, thus, 



if 00 if 00 



- cos svt ds = - 3o(sr) cos svt ds , (177) 



^o ^o 



the value of which is known to be (175). Conversely, we may 

 evaluate the definite integral by turning it to the analytical form 

 gT (gr), which may be done by inspection, and then integrating 

 through the equivalent operator H (gr)-Jg. But this definite in- 

 tegral is only one of several that may be immediately derived from 

 the operators in (173), (174) by harmonic decomposition, and it 

 will be more convenient to consider them separately in later sec- 

 tions along with applications and extensions of the preceding. 



