J'tdy 1 1, 1S89] 



NATURE 



247 



boulder clay, and numerous examples of striated rock surfaces 

 and oiher phenomena occurring below the low-level marine 

 boulder clay can be quoted in support thereof. 



T. Mellard Reade. 

 Park Corner, Blundellsands, June 5. 



Test of Divisibility by any Prime. 



In Nature of ]May 30 (p. 115), Mr. Tucker has given the 

 formula : — 



N = 21M + io''-i(7Q) = 7Q'. 



In an exactly similar way we may show that — 



N = iiM + 10""^ (llQ) = liQ', giving a multiplier I, 

 N = 91M + ic»-i(i3Q) = 13Q', „ „ 9, 



N = 51M + ic"-J (17Q) = 17Q', „ „ 5, 



<S;c., &c., for any number ending in digits i, 3, 7, or 9.^ 



The general principle may be simply shown as follows : — 



We have 17x3 = 5T, say. 



This means (i) that, if any number ends in unity, and is also 

 of Form 17M, then all the figures to the left of unity will form a 

 number of Form 17M + 5. 



It also means (2) that, if we multiply the units digit by 5 (casting 

 out the prime 17, if need be), we get the figures to the left ; e.g. 

 2346 ends in 6, and is of Form 17M. Therefore, 234 is of 

 Form 17M + 13 (since 17x8= I3'6) ; also, 6 x 5 = 30, and 

 30- 17= 13. The process can be repeated to any extent. 

 Thus, since 234 = 17M + 13, subtract 13 from 234, giving 

 221 = l7Mj. Since 221 ends in unity, therefore 22 is of Form 

 17M + 5, and, subtracting 5 from 22, we have 17 = 17M2. 

 Hence the rule. 



From similar considerations I have deduced the following 



formula, giving the periodicity of -^ where N is a prime : — 



If [{(7<N + O/ic}/ + N - i]/N = I (an integer), then/ will 

 be the periodicity of i/N. 



Here u means the unit's digit of N, or else the integral 

 quotient of 9 divided by the unit's digit. 



Thus for all numbers ending in 9 the formula becomes 

 [{(N + \)l\oY + N - i]/N, e.g. {2P + i8)/i9 gives the period- 

 icity of 1/19, &c. 



The corresponding formulse for num^;ers ending in 7, 3, i, 

 are, respectively, 



[{(7N + i)/ioi/ + N - i]/N ; [{(3N + i)lioY + N - i]/X ; 

 [!(9N -f i)/io}/ + N - iJ/N. 



Another useful deduction from the same principle is : — 



If/ be the periodicity of the recurring fraction i/iV (where N 

 ends in r, 3, 7, or 9\ then the test will give the true remainder 

 of any / + 2 figures ; e.g. What is the remainder of 98765 -^ 37 ? 



Since 37 x 3 = iit, our multiplier is II. 



Therelorc ^876 - II x 5 = 9821, and 982 - il x i = 971. 



Also 97 - II X I = 86 = 37M + 12. Thus 12 is the 

 remainder. 



I find that by this new process the remainder may be obtained 

 in about one-half of the time taken by the ordinary method of 

 division. Roirr. W. D. Christie. 



Wavertree Park College, Liverpool. 



QUARTZ FIBRES? 



T N almost all investigations which the physicist carries 

 ■*■ out in the laboratory, he has to deal with and to 

 measure with accuracy those subtle and to our senses 

 inappreciable forces to which the so-called laws of Nature 

 give rise, Wheiher he is observing by an electrometer 

 the behaviour of electricity at rest, or by a galvanometer 

 the action of electricity in motion ; whether in the tube of 

 Crookes he is investigating the power of radiant matter, 

 or with the famous experiment of Cavendish he is finding 

 the mass of the earth— in these and in a host of other cases 

 he is bound to measure with certainty and accuracy forces 

 so small that in no ordinary way could their existence be 



' These numbers only can give, when multiplied, all the digits in the units 

 place. 



"Lecture delivered at the Royal Institution, on Fiiday, June 14, by Mr. 

 C. V. Boys, F.R.S. 



detected ; while disturbing causes which might seem to 

 be of no particular consequence must be eliminated if his 

 experiments are to have any value. It is not too much 

 to say that the very existence of the physicist depends 

 upon the power which he possesses of producing at will 

 and by artificial means forces against which he balances 

 those that he wishes to measure. 



I had better perhaps at once indicate in a general way 

 the magnitude of the forces with which we have to deal. 



The weight of a single grain is not to our senses appre- 

 ciable, while the weight of a ton is sufficient to crush the 

 life out of anyone in a moment. A ton is about 15,000,000 

 grains. It is quite possible to measure with unfailing 

 accuracy forces which bear the same relation to the 

 weight of a grain that a grain bears to a ton. 



To show how the torsion of wires or threads is made 

 use of in measuring forces, I have arranged what I can 

 hardly dignify by the nameof an experiment. It is simply 

 a straw hung horizontally by a piece of wire. Resting on 

 the straw is a fragment of sheet-iron weighing ten grains. 

 A magnet so weak that it cannot lift the iron yet is able 

 to pull the straw round through an angle so great that 

 the existence of the feeble attraction is evident to every 

 one in the room. 



Now it is clear that if, instead of a straw moving over 

 the table simply, we had here an arm in a glass case and 

 a mirror to read the motion of the arm, it would be easy 

 to observe a movement a hundred or a thousand times 

 less than that just produced, and therefore to measure a 

 force a hundred or a thousand times less than that exerted 

 by this feeble magnet. 



Again, if instead of wire as thick as an ordinary pin I 

 had used the finest wire that can be obtained, it would 

 have opposed the movement of the straw with a far less 

 force. \^ is possible to obtain wire ten times finer 

 than this stubborn material, but wire ten times finer is 

 much more than ten times more easily twisted. It is ten 

 thousand times more easily twisted. This is because 

 the torsion varies as the fourth povver of the diameter, so 

 we say 10 X 10 = 100 ; 100 x 100 = 10,000. Therefore 

 with the finest wire, forces r 0,000 times feebler still could 

 be observed. 



It is therefore evident how great is the advantage of 

 reducing the size of a torsion wire. Even if it is only 

 halved the torsion is reduced sixteen-fold. To give a 

 better idea of the actual sizes of such wires and fibres as 

 are in use I shall show upon the screen a series of 

 photographs taken by Mr. Chapman, on each of which a 

 scale of thousandths of an inch has been printed. 



The first photograph (Fig. i) is an ordinary hair — a suffi- 

 ciently familiar object, and one that is generally spoken of 

 as if it were rather fine. Much finer than this is the specimen 

 of copper wire now on the screen (Fig. 2), which I recently 

 obtained from Messrs. Nalder Brothers. It is only a little 

 over one-thousandth of an inch in diameter. Ordinary 

 spun glass, a most beautiful material, is about one- 

 thousandth of an inch in diameter, and this would appear 

 to be an ideal torsion thread (Fig. 3). Owing to its fine- 

 ness its torsion would be extremely small, and the more so 

 because glass is more easily deformed than metals. Owing 

 to its very great strength, it can carry heavier loads than 

 would be expected of it. I imagine many physicists 

 must have turned to this material in their endeavour to 

 find a really delicate torsion thread. I have so turned 

 only to be disappointed. It has every good quality but 

 one, and that is its imperfect elasticity. For instance, a 

 mirror hung by a piece of spun glass is casting an image 

 of a spot of light on the scale. If I turn the mirror, by 

 means of a fork, twice to trhe right, and then turn it back 

 again, the light does not come back to its old point of 

 rest, but oscillates about a point on one side, which, how- 

 ever, is slowly changing, so that it is impossible to say what 

 the point of rest really is. P'urther, if the glass is twisted 

 one way first, and then the other way, the point of rest 



