242 Mr. Blackburn on the Modem Telegraphs. 



Let AB be an upright pole or staff, carrying any number 

 of arms FC, DG, HE, &c., moveable in a 

 vertical plane about the centres F, D, E, &c, ^ ^> 



and capable of being placed in any number fb^ c 



of positions. This will represent the modern 

 telegraph or semaphore. 



Let the number of centres F, D, E, &c. 

 be denoted by c, and the number of positions 

 of each arm by p. Then, since the number 

 of signals that can be made by one arm must 

 be equal to the number of positions of that 

 arm, the number of signals which can be 

 made with one arm = p 9 and the number of 

 signals on the whole, using one arm at a 

 time, will be cp. 



Again, since each of the signals which can 

 be made by any one arm, can be repeated with each of the 

 signals of any other arm, it follows that the number which 

 can be made by any two arms together = p % . And since 

 the number of combinations in c things, taken two and two 



together, = ° \ C ~ , it follows that the whole number of sig- 



EI 



PB 

 B 



1 .2 

 nals, using two arms at once, = 



c.c—l 

 1 .2 



In like manner it may be shown that the num ber of sign als 



c . c-~— 1 . c mmm 2 

 which can be made using three arms at once = ^— — 5— — p 3 ; 



and that when all the arms are used at once, the number will 



be 



c.c—] .c-2 



c-(c-l) 



p c 9 the index of p being al- 

 1.2.3 c 



ways = number of arms used at once. The whole number of 



signals, therefore, which the telegraph is capable of making 



will be cp~^ 



. c.c—l 



1 .2 



c.c—l ,c — 2 



J 



2. J 

 &c. 



+ 



c.c- 



+ &c. 

 c— 2 c— (c 



1.2.3 



But if the binomial 1 +p be raised to the cth power, it will 

 be found to coincide exactly with the sum of the preceding 



