72 BELL SYSTEM TECHNICAL JOURNAL 



of p are required at the difl"erent roots, as shown for ( — 1 in the table 

 above. A logical extension would therefore be to make p a function of .v 

 such that it takes on the required values at ri , r-j , rs , • • • . When this is 

 done and p({x) is introduced into (1) and (2), one has to integrate 



/ 



K.v)/"(..-) ,,^ 



and this is intractable. 



Hence p{x) is made a discontinuous function, such that p has the value 

 pi corresponding to ;'i for values of .v from zero to a point bi between ri and 

 r-i ; the value p2 corresponding to r^ for values of .v from bi to a point bi be- 

 tween r-i and rs; and so forth. This introduces discontinuities in </>. No 

 discontinuities exist, however, in the function 



G( = e~'( (9) 



which is given in Table II. The calculations were made by Miss F. C. 

 Larkey; numerical integration was according to Weddle's rule. 



Within the limits of this tabulation, then, G( and F( are now considered 

 to be known functions. 



Table I 

 Power Series Expansions of 4>t{x) 



/ ^p\ /I 17A /7 19p\ 



,,,,, , (■ , _ _f j , + (^^ _ -^ j ,. + (^- - _ j ...+ .,. 



= -0.063813.V -0.001 178x3 -0.0000358.v5 _ ... 

 *,W - - ^) .V + (i - '^^ .V. + [^ - ^^ -V + . . . 



= +0.15451.V +0.01648.r' - O.OO.SSO.v^ - ••■ 



/! Sp\ ( \ 41/. \ / 13 103/> \ 



'^^^■^' = (i - 2ij -^ + Vn - 5760 j -^"^ + (,17280 " 276480 j "^ + 



= +0.12210.V +0.00667.V' +0.00375.vS - ••• . 



^ Unless p = b + cJ' {b and c constants), which is not of any use. 



