43 



the remainder may be written I3(a + b + c), which is divisible by 13, 

 whatever be the value of a, b and c. 



In the same way it may be shown that a number of this kind is 

 divisible by 11. 



When the first figure of the period is 0, and the second any what- 

 ever i and./, the number is OijOij=ijOij; or any number of five places, 

 the first two and the last two being the same, while the middle place 

 is 0, is divisible by 7, 11 and 13. Thus 34034, 14014 are so 

 divisible. 



When the first two places are 0, the number may be written 

 00z'00z = z00/, or any number of four places, the first and last figures 

 being the same, while the two middle places are 0, is divisible by 

 7, 11 and 13. Thus 5005, 8008 are so divisible. 



Like properties may be found for 17, 19, 23, but the periods are 

 longer. The prime divisor being In + 1 , it is manifest the number 

 of places in the period cannot exceed, however it may fall short of . 



Thus when the divisor is 17, the number of places in the period is 

 eight. 



III. " On Fessel's Gyroscope." By C. WHEATSTONE, Esq., 

 F.R.S. Received April 6, 1854. 



Since the announcement of M. Foucault's beautiful experiment 

 which has afforded us a new mechanical proof of the rotation of the 

 earth on its axis, the phenomena of rotary motion have received re- 

 newed attention, and many ingenious instruments have been con- 

 trived to exhibit and to explain them. One of the most instructive 

 of these is the Gyroscope invented by M. Fessel of Cologne, de- 

 scribed in its earlier form in Poggendorff's Annalen for September 

 1853, and which, with some improvements by Prof. Pliicker and 

 some further modifications suggested by myself, I take the present 

 opportunity of bringing before the Royal Society. 



It is thus constructed : a beam is capable of moving freely round 

 a horizontal axis which is itself moveable round a vertical axis, so 

 that the beam may move in any direction round a fixed point ; at 

 one end of the beam is fixed a horizontal ring which carries a heavy 



F2 



