366 Mr. Blackburn on the Modem Telegraphs, 



T) T) — 1 T) ■" ■" 2 



when three arms are made use of, will be - : — — — '— ; and 



when all the arms on the centre are used, the signals will be 



\ l£. *" s£ — - . Hence the total number of signals 



1 .2 a-1 .a 



which can be made by the arms on a single centre will be re- 

 presented by the sum of the series 



^+-172" 1.2.3 + ' &C 1.2 T=l.«~ 



to a terms. 



Let the sum of this series be denoted by A ; then since the 

 number of centres = c, the whole number of signals which can 

 be made, using the arms on a single centre at a time, =cA. 



Again, since the signals which can be made on any one 

 centre can be combined with each of the signals on any other 

 centre, it follows that the number which can be made, using 

 any two centres at a time, = A 2 . 



Also, since the signals which can be made upon any two 

 centres may be combined with each of the signals upon any 

 other centre, it follows that the number of signals, using any 

 three centres at once, will be A 3 . 



In like manner it may be shown that the signals, using c 

 centres at once, will be A c . Hence 



The number of signals using one centre at a time as A 



— two centres = A 2 



&c. &c. &c. 



c centres = A c 



Again, since the number of combinations of c things, taken 



c . c 1 



two and two together, = ~- — — -, it follows that the number of 



different pairs of centres which can be taken together will be 



* ~ ; and since the number of signals with each pair of 



centres = A 2 , the total number of signals which can be made, 



c • c —~ 1 

 using a pair of centres at once, will be —J — — - . A 2 . 



1.2 



Also, since the number of combinations of c things, taken 

 three and three together, is — '— — ^—- — , the number of dif- 



