TRANSIENT SOLUTIONS OF ELECTRICAL N EI WORKS 133 



Appendix 

 1 



The expression 



R^ \ I — g-2(i?//so+P) 



can be expanded into the form 



1 



4R 



Ro\l- g-2(iJ/i?o+P) 



— tn 



- 1 



1 



Ro \i - g-2(«/if0+F) 



+ 



+ (- l)'^ (55) 



and hence the general solution depends only on the solution of the 

 general form 



*^^ • ^T" (56) 



Ro \l - g-2(«/iJo+P) 



If equation (56) is expanded by the binomial theorem, there results 

 the expression 



Uo/ 



I _l_ ;;2g-2{fl//eo+F) -|- ^!ii!!LlL22 g-4(ij/Ko+p) 



2! 



+ 



+ 



(m -\- n - i)!g-2n(fi/«o+p) 



n\{m — 1) ! 

 For any value of m, we can write the typical term of (57) as 



m -\- n — I \'»+«-i 

 (m + w — 1) ! 



(57) 



■\2Tr{m + w — 1) 



n\{m — 1 ) ! 



llivnim — 1) ! 



g-(m-l) 



y ■ j im + n- 1)' 



-1/2 



(w — 1) \-\n 



by Stirling's theorem on factorials. Now w for any finite value of time 

 approaches infinity, while 7n for any finite term in the series is finite. 

 Hence (58) can be written as 



g-(m-l) 1 ^ 



m — \ 



{m - 1) ! 



77^ — 1 \ ^ 



The limit of ( 1 + ) as w -> <» is ^('"-D.e Hence 



{m -\- n — 1) ! 



n\{m - \)\ {m - 1)! 

 ^ See "Probability and Its Engineering Uses," T. C. Fry, page 107. 



