744 BELL SYSTEM TECHNICAL JOURNAL 



Theory, Vol. II, shows that they may be divided into two classes: 

 (I) those of which the operational equation is of the form 



h = F{pWp (I) 



and (II) those of whirh the operational equation is of the form 



h=4><pWp) (ID 



where k is an integer. 



Heaviside himself does not (lislinj4ui>!i liftween tlie two classes, 

 but employs the following rule for obtaining as\niptotic expansion 

 solutions : 



If the operational equation 



h = l/H{p) 



can be expanded in the form 



h=ao+aip+a<p-+ . . +a„p"+ . . . 



{bo + b>p + b-2p'+ . . +b„p"+ . . . )Vp, (107) 



a solution, usually divergent, is obtained by discarding the first expansion 

 entirely, except for the leading constant terms «„, replacing \/p by \/\/irt 

 and p" by d" jdt" in the second expansion, whence an explicit series 

 solution results. 



It should be expressly understood that Heaviside nowhere himself 

 states this rule formally. He does not distinguish between the two 

 cases where integral series in p do and do not appear, although very 

 important mathematical distinctions are involved. Furthermore, 

 in one case he modifies his usual procedure b\- adding an extra term 

 (Elm. Th. Vol. II, pg. 42-44). It certainly represents, however, 

 his usual procedure in a very large number of proi)len)s. 



A completely satisfactory theory of the Heaviside Rule, just slated, 

 has not yet been arri\ed at although we can ahva>s verify the di\er- 

 gent solutions in specific problems. Furthermore, it is not as yet 

 known just how general it is, though it certainly works successfully 

 in a large number of physical problems to which it has been applied. 

 Finally we know nothing in general as to the asymptotic character 

 of the resulting expansion. In some rases it leads to an ex|ian.sion 

 in which the error is less tli.m the last term included, in otiiers re- 



