Mr. Blackburn on the Modem Telegraphs. 243 



terms, with the exception of unity, by which it exceeds it. 

 The expression (l+p) c — 1, therefore, will accurately repre- 

 sent the whole number of signals. 



Example. — In the Admiralty semaphore, the number of 

 centres or arms = 2, and the number of positions of each 

 arm = 6. Hence, by the theorem, the number of signals 

 will be (6 + l) 2 -l = 7 2 -l = 48. 



Example 2. — If the number of centres be three, as in the 

 preceding sketch, and the number of positions of each arm be 

 six, the number of signals will be (6 4- 1 ) 3 — 1 = 7 3 — 1 = 342. 



Example 3. — In the year 1796, a line of telegraphs was 

 established between Paris and Landau, each of which had 

 seven arms, and each arm seven positions. It is required to 

 find the number of signals. 



By the theorem, the signals m (7+1) 7 — 1 = 8 7 — 1 = 

 2,097,151. 



Corollary. — From the preceding investigation it appears 

 that, for any semaphore having c centres, and each arm p posi- 

 tions, the number of signals which can be made, using one 

 arm at once, will be represented by the second term of the bi- 

 nomial 1 +p raised to the cth power ; the number of signals 

 using two arms at once, by the third term of the same quan- 

 tity, &c; and the number of signals, using all the arms at once, 

 by the (c+l)th term. 



Thus, in the Admiralty semaphore, 

 The signals using one arm at once = cp = 2x6=12 



c c — 1 

 The signals using two arms at once = ff 9 = 1 x 6 9 = 36 



Total number of signals, as before, = 48 



In the Paris semaphore before mentioned, the signals using 



one arm at once = cp = 7x7 = 49 



two arms at once = ° \ C ~ X p* = 21x49 = 1029 



1.2 ' 



three arms at once = c -L£IzLl£Z^ p s m 35x343 = 12005 



1.2 .0 



four arms = c ' c ~ l - g -~ 2 -- g - 3 p < — 35x2401 = 84035 



1.2.3.4 * 



five arms ... « c ' c ~ l ' C ~ 2 ' C ~ 3 ,c 7 4 P> = 21 X 16807=352947 



1 . x, . o . 4 . o * 



c.c — l.c — 2.c — 3.c — 4.c — 5 « „ .,«.««* #»<*»»** 

 six arms = ^ g 3 ^ ^ g—p 6 = 7x 117649=823543 



scven= r.7\'7 2 'T 3 'T 4, T 5 ;7 6 ^ 7==lx823543 =823543 



Total number of signals, as before, = 2,097,151 

 2 12 



