30 



NA TURB 



[November i i, 1897 



LETTERS TO THE EDITOR 



[ The Editor does not Jiold himself responsible for opinions ex- 

 pressed by his correspondents. Neither can he undertake 

 to return., or to correspond with the writers of., rejected 

 vianiiscripis i77tended for this or any other part of Nature. 

 No notice is taken of anonymous romtnunications.'l 



The Law of Divisibility. 



May I briefly supplement my former letter by a few sugges- 

 tions for the development of the above law ? 



(i) When 5 (or a multiple) appears; in N, it may be replaced 

 by cyphers. 



Thus if 5 = 3, for 235697 write 205007, 

 = 7, do. 200690 1 



or 230090 j 

 = 23, do. 5007. 



(2) Any member of the recurring period r„ may be repre- 

 sented by its negative complement which reaches its maximum 

 at ^5. 



Thus if 5 = 7, period = ± i, 3, 2 ; 



= II, do. = I, - I ; 



= 37, do. = I, 10, - II ; 



r= 41, do. = I, 10, 18, 16, - 3. 



(3) If the final remainder be negative, its complement must 

 be taken. 



Thus if 5 = 7, 17, 29, 41, 



and R = - 2, - 14, - 25, - 32, 

 true value = 5, 3, 4, 9. 



{4) Final remainders may be found by repeated applications 

 of the requisite formula. 



Let 5 =41, 



N = 3205175 or 3000175 



Nj = 5 + 70 + 18 + 30 



= 123 = 3 + 20 + 18 = 41, 

 Let 5 = 37, 



N =87172 



Nj - 2 + 70 - II + 7 + 80 



= 159 - II = 148 = 8 + 40 - II = 37. 

 Let 5=7, 



N = 8638 



Ni = 8 + 9+ 12 -8 = 21 = 1 + 6 = 7. 



(5) The group principle may be applied to 5 = 99, 999, 9999 

 &c., where Nj = a^ + b^ + Sec. ; a<f + b^ + &c. ; a^ + b^ + &c. 

 The first is a test for 11, the second for 37, the third for loi. 



(6) Another method is the following : — 



Let N 4- 5 + I = Q, with remainder r^, 

 Qj ^ 5 + I = Q2 do. ^2, 



Q2 -^ S + I = Qs do. ^3, 



&c. = &c. 



.-. N = (8 + !)Q, + r, 



= (5 + i)2Q., + (S + i)r^-\-r, 



= (5 + l)'Q3 + (5 + 1)^3 + {S+ l)r.,-\- ;-! 



= Sec. 



jiliminating multiples of 5, we get, when Q^ = o, 



Nj = r„.i + r„-2 + &c. + r^. 



If 5 ± a be used, we get 



Nj = a»-Jr„_i ± a "" V„_2 + &c. ± r^. 



Putting a = I, 2, 3 we may deal with a wide series of primes, 

 such as 



19, 29, 59, 79, 89, 109 &c. 



31, 41, 61, 71 &c. 



23. 43. 53. 73, 83 &c. ; 



also with composites, as 



119 for 17 and 7, 129 for 43 and 3, 

 159 for 53 and 3, 201 for 67 and 3, 

 301 for 43 and 7, 501 for 167 and 3, and so on. 



As examples, let S = 399 = 3 x 7 x 19. 



N = 8293177893 



NO. 1463, VOL. 57] 



399 = 19 X 7 X 3. 

 Let S = 299 = 13 X 23, 

 N = 166371972 



-^ 30oS r^ + r2 + r^+ r^ = 72 + 173 + 48 + 6 = 299. 



Let 5 = 501 = 167 X 3 

 N == 640550043 



-r- 500*, 5 - 62 + 100 - 43 = o. 



From the foregoing I have derived many simple rules no 

 requiring division. Henry T. Burgess. 



Tarporley, West Norwood, November 4. 



A Link in the Evolution of a Certain Form of Induction 

 Coil. 



At a time when much interest is taken in the oscillatory 

 electric discharge and its effects, it may not be out of place to 

 mention that a link in the evolution of the Tesla coil is to be 

 found in a paper by Dove (Royal Academy of Sciences, Berlin, 

 October, 1844 ; Electrical Magazine., vol. ii. p. 67). It is as 

 follows : — The external coatings of two Leyden jars were con- 

 nected together by a wire spiral. This spiral was surrounded 

 by a secondary insulated spiral. When the jars were so charged 

 that a spark was produced on joining their internal coatings, 

 electricity was induced in the secondary spiral. If to this 

 arrangement of Dove, a cistern of insulating oil be added to 

 contain the coils, and the jars, furnished with a spark gap, be 

 charged from an induction coil, we have one of the combinations 

 which has given such excellent results in the hands of Tesla. In 

 1831 Faraday (" Experimental Researches," vol. i. § 24) arranged 

 an experiment to discover whether the electrical discharge of a 

 Leyden jar would produce an induced current in his induction 

 coil ; he writes : " Attempts to obtain similar effects by the use 

 of wires conveying ordinary electricity (i.e. from a jar) were 

 doubtful in results." 



The combination due to Dove, is probably the earliest instance 

 of an apparatus in which electrical oscillation in one circuit 

 set up a definite disturbance in a neighbouring coil. 



Oxford, November 8. F. J. Jervis-Smith. 



The Leonid Meteors. 



I SHOULD be glad to receive accounts of any brilliant meteors 

 that may be observed on the nights of November 13 and 14 next, 

 for the purpose of computing their real paths in the air. The 

 date and time of appearance of each object should be given, 

 together with its apparent magnitude (compared with the moon, 

 planets or brighter stars), observed course amongst the stars in 

 R.A. and Declination, and estimated duration of flight. Though 

 moonlight will be strong, many observers will be on the look-out 

 for the vanguard of the Leonids, so that should any brilliant 

 meteors appear, they are likely to be noticed at several different 

 stations. W. F. Denning. 



51 Brynland Avenue, Bishopston, Bristol. 



Insects and Colour. 



The following incident may throw some further light on the 

 subject brought forward by your correspondent, Mr. J- Parkin, 

 in his letter in your issue of November 4, on " A Bee's 

 Movements in a Room." In the year 1893, the humming-bird 

 hawk moth was particularly common here. On one or two 

 occasions, driving out in a little trap, with a Shetland pony, whose 

 head-gear was ornamented with pyramidal blue rosettes, one of 

 these beautiful insects would fly straight at one of the rosettes, 

 and hover over it for a few seconds, though the pony was going 

 at a trot. It would seem that in this case the colour alone was 

 the chief attraction ; the odour being insignificant. But there 

 are, I believe, numerous other instances of insects being attracted 

 in the first instance by colour. I may add that these insects 

 visited chiefly the scarlet geraniums in my garden. 



Alfred Thornley. 



South Leverton Vicarage, Notts., November 5. 



