523 Royal Society. 



any three figures, is divisible by the prime numbers 7, 11 and 13. 

 Thus 376376, 459459, 301301 are so divisible. 

 A number N of six places may be thus written : — 



N=100.000a+ 10.000 6+ 1000 c+ 100 c?+ 10 e+/, 



which, when divided by 7, will give a quotient q and a remainder 

 5a + Ab+&c+-2d+Ze+f. 



Now if d=a, e—h,f=iC, this remainder may be written 7(« + 6 + c), 

 which is divisible by 7, whatever be the values of a, b, c. 



In like manner if a number of six places be divided by 13, the 

 remainder will be 



4a + 36+12c+9(^+10e-|-/; and, as before, if d=a, e=b,f=c, 

 the remainder may be written 13(a + 6 + 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 andy, 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, 1 1 and 13. Thus 34034, 14014 are so 

 divisible. 



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

 00i00i=i00(, 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 2n + l, it is manifest the number 



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



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



eight. 



3. " On Fessel's Gyroscope." By C. Wheatstone, Esq., F.R.S. 

 Since the announcement of M. Foucault's beautiful experiment 

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

 earth on its axis, the phsenomena 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 

 disc, the axis of rotation of which is in a line with the beam ; at the 

 opposite extremity is a shifting weight by means of which the equi- 

 hbrium of the beam may be established or disturbed at pleasure. 



If the beam be brought into equilibrium, and the disc be rapidly 

 rotated, by means of a thread unrolled from its axis, it will be seen 



