Mr. Blackburn on the Modern Telegraphs, 367 

 ferent sets of three centres will be C -^fl^l ; and since the 



number of signals which can be made upon any three centres 

 is A 3 , it follows that the total number of signals, when three 



centres are used at once, will be c ' c ~ l *J^g as &c 



1.2.3 ' 



In like manner, the numbers, when c centres are used at 



once, will be C *"- ri "' *l£=i fe^Ej} a* 



1.2 *~l).c 



It appears, then, that the whole number of signals which can 

 be made by the instrument, will be represented by the sum of 

 the series 



cA + c -^=La»+,&c c.<T=T ... {c-TZTi} Ac 



1 * A 1.2 c 



to c terms, in which expression A = p+ P ' P ~~ ■ -f, &c. to a 

 terms. But by the binomial theorem, the sum of the series, 



cA+ c 4^ 1 A*+&c. c - MB "{'-^1 A . 



1 • ^ 1.2 c 



I - (A+l)«-l; 

 hence the number of signals will be represented by the for- 

 mula (A + l) c — 1. 



Example. — Let it be required to find the number of signals 

 which can be made by the Admiralty semaphore. 



In this example a = 1, c = 2, /> = 6; and since a = 1, 

 A =x a, and the expression (A + l) c — 1 becomes (6+ I) 2 — 1 

 = 48, the number required. 



Example 2. — To find the number of signals which could 

 be made by the Admiralty semaphore supposing it to have 

 two arms upon a centre. 



Here a = 2, .*. A = p • —q— = ~k~ » nence tne signals 

 willbe(15+l) 9 -l = 255. 



Corollary. — The number of signals which can be made, 

 using any one centre, will be represented by the second term 

 of the binomial A + 1 raised to the cth power ; the number 

 using two centres, by the third term of the same power; and 

 so on. Thus, 



