KENNELLY. — ARTIFICIAL LINES FOR CONTINUOUS CURRENTS. 129 



Appendix. 



The demonstrations of the various formulas appearing in the fore- 

 going paper have been omitted in order to save space. Nearly all of 

 these formulas are, however, based upon and derived from the follow- 

 ing propositions : 



(1) Any alternating continued fraction is expressible as a constant 

 continued fraction. Thus to n stages : 



1 b 1 



= — ?=■ X 



a + 1 Vab Vab + 1 



b + 1 Vab + 1 



a + 1 Vab + 1 



b + Vab + 



(2) Any constant continued fraction is expressible as a simple single 

 fraction or ratio of a hyperbolic sine and cosine. Thus the nth con- 

 vergent of 



1 sinh nO . . 



— T~^ = — r~? — , -.n n " n ls even » or 



c + 1 cosh (n + 1) 6 



c+1 



c 



cosh nO .. . , . 

 it n is odd, 



sinh (n+ 1) 



where 6 = sinh -1 ( °- 



(3) Any terminally loaded constant-continued fraction is expressible 

 as a simple fraction or ratio of hyperbolic sine and cosine. Thus the 

 nth ascending convergent of 



sinh (nO + 0) 



c + 1 cosh [(n + 1) 6 + <f>] 



c+ 1 



c + m 



cosh (nd + (j>) 



sinh[0i +1)0 + 0] 



if n is even ; or 



if n is odd ; 



where $ is an auxiliary hyperbolic angle. 



(4) The sending-end resistance of any artificial line composed of sim- 

 ilar sections, whether the leaks are in the middle or not, may always 

 be expressed as a terminally loaded alternating continued fraction. 

 vol. xliv. — 9 



