74 



KNOWLEDGE. 



[April 1, 1893. 



short wave-lengths beyond the violet end of the spectrum, 

 which affect the photogi'aphic plate but do not affect the 

 eye ; consequently the photographic trace left by Sirian 

 stars is denser than that left by solar stars of the same 

 \'isual magnitude. Prof. Pickering has shown from an 

 examination of the stars contained in the Draper Catalogue 

 (see Annuls of Harrurd Coll. Ois., vol. xxvi., p. 1.52| that 

 stars of the second and third type above the G^ magnitude 

 are distributed nearly equally over the northern heavens, 

 while stars of the A or Sirian type above the 6i magnitude 

 are nearly twice as numerous in the region of the Milky 

 Way as in the rest of the sky. Mr. Marth, in his list of 

 galactic stars, does not profess to give all stars down to the 

 6th magnitude. His stars are chosen so as to give 

 convenient points of reference for drawing the Milky Way. 

 Mr. Proctor's discussion in the Monthh/ Xoticrs twenty 

 years ago, as to the probability of the observed number of i 

 stars brighter than the 6th magnitude falling by chance on 

 the area of the Milky Way, seems to me to prove con- 

 clusively that the majority of these stars must be actually 

 associated with the Milky Way. But if this class of 

 evidence does not appeal to Mr. Monck's mind, I should 

 have thought that an examination of the recent photographs 

 of the Milky Way could have left no reasonable doubt that 

 many of the brighter stars in the region of the heavens 

 occupied by the Milky Way are intimately associated with 

 the regions of nebulosity on which they are projected, and 

 that, consequently, these stars are at the same distance 

 from us as the nebulosity. — A. C. Ranyakd.] 



Let ffj, a„, &o., be the digits taken in order of any 

 number, N,"rtj, being the digit in the units' place. Then 

 (scale ;■) 

 N and </, + ti2 + a^+ . . . leave the same remainder when 



divided by r — 1 ; and 

 N and «j — aj + rtg- . . . leave the same remainder when 



divided by r+1 

 with the proviso in the latter case that , when the sum is nega- 

 tive, the true remainder is found by subtracting the actual 

 remainder from ;+l. These two statements are proved 

 in most algebras and, when the scale is 10, constitute 

 the usual rules for divisibility by 9 and 11 respectively. 

 We can at once deduce other results by simply observing 

 that the above statements hold when Oj, a.^, &c., represent, 

 instead of single digits, sets of any the same number of 

 consecutive digits, r -1 becoming the same number of 

 digits, each equal to ;— 1 ; and a similar change being 

 made in the meaning of ;■+!. In fact, the latter ex- 

 pression becomes, whatever the radix, 10 01, the 



number of digits being one more than the number in each 

 of the sets. Thus, in scale of 10 : 



(1) 99 = 9 X 11 ; .-. 977G547 is divisible by 9 and 11, if 

 47 -f 65 + 77-^9 = 198; /.-■., if 98-1-1 = 99 is. 



THE INSECT PLAOUES OF URUaUAY. 

 To the Editor of Knowledge. 



Dear Sir, — The pastoral country of Uruguay depends 

 for its prosperity upon the amount of rainfall. For the 

 last four or five years the drought has been exceptional ; 

 occasional slight showers which have fallen seem to do 

 more harm than good, as they merely moisten the surface 

 without penetrating to the roots, and the strong sun 

 immediately cakes the ground again. Since August, 1891, 

 there has not been enough rain to fill the rivers and 

 streams. In consequence, the sheep and cattle have been 

 unable to find any sustenance in the shrivelled grass. But 

 though the higher animals have been reduced in numbers 

 by famine and disease, the unnatural dryness has favoured 

 the increase of insect life. 



A little grey beetle named Vaquilla has swept over the 

 land in swarms, devouring every remnant of vegetable life. 

 It seems to be something akin to the well-known Colorado 

 beetle. Amidst the general sterility it seems to flourish. 



A more curious, or at any rate less known beetle, is the , 

 Isoca. Not content with devouring the surface produce, it 

 begins its work underground. The grub is like a fat white 

 worm, about two inches in length, and half an inch in 

 diameter. It burrows in the ground, cutting off the roots 

 of every vegetable it comes across — wheat, maize, and 

 other cereals, as well as grass. At intervals it throws up 

 mounds of earth on the surface, which are like diminutive 

 mole-hills. Sometimes these so completely cover the 

 ground that it is impossible to step between them. 



Yours faithfully 



G. E. MiTTON. 



To the Editor of Knowledge. 



Sir,— The following simple tests of the divisibiHty of 

 any number by 7, 1.3, and other primes, seem wortliy of 

 record. They are not given in any algebra with which I 

 am acquainted, but of course very possibly they are well 

 known. 



Again, 75432141 is so, if 41-f-21 -1-43 + 75 = 180 ; 



if 



80+1 = 81 is so. Hence the latter number is divisible by 

 9, and leaves remainder 4 when divided by 11. This 

 method seems shorter than the application of the two 

 ordinary rules: 7 + 5 + 4 + 3 + 2 + 1+4-1-1=27; 1 + 1 + 3 

 + 5 = 10; 4-1-2 + 4 + 7 = 17; 10-17=-7; 11-7=4. 



(2.) 999=27 X 37 ; 357202995 



995 + 202 + 357=1554; 554 + 1=555; .-. the number 

 is di\dsible by 37 ; and, when divided by 27, the remainder 

 is 15. 



(3.) 9999=9x11x101; and we can proceed as before 

 but with sets of four digits, And so on. 



(4.) 88529631 ; 31-96 + 52- 88= — 101 ; .-.the number 

 is divisible by 101. 



(5.) 1001=7x11x13; 71420843; 843-420 + 71 = 494; 

 .-. the number is divisible by 13, and leaves remainders 10 

 and 4, when divided by 11 and 7 respectively. 



(6.) 10001 = 73x137, which gives us easy tests for the 

 primes 73 and 137. And so on. 



(7.) IfP = rti-«4+"7- . . .■,Q = a„-a^+a^- . . .; 

 and E—a^-a^+a^- . . . .; then N and P + 3Q + 2R 

 leave the same remainder when divided by 7 ; and N and P 

 — 3Q — 4R leave the same remainder when divided by 13, 

 with the same proviso, when the sum is negative, as in the 

 ordinary rule for divisibility by 1 1 . 



Example, 54269761. Here 1 -9 + 4= -4 ; 6-6 + 5 = 5; 

 and 7-2=5; .-.-4 + 3 5 + 2.5=21; and - 4-3.5- 4.5 

 = — 39. Hence the number is divisible by 7 and 13. 



(8.) N and («,+",+ • • .) + 10 («.,+«,+ . . .)-ll 

 (".3 +"0+ • • •) leave the same remainder when divided 

 by 37, with proviso as before in regard to negative sum. 



Example: 145989901; 1+9 + 5 = 15; + 8 + 4 = 12; 

 9 + 9 + 1 = 19; 15 + 10.12-11.19= -74. Hence the 

 number is divisible by 37. 



Hereford, March 2nd, 1893. Edwyn Anthony. 



To the Editor of Knowledge. 



Sir, — While trying to find a formula for the solution of 

 " figure squares," similar to the one which I give below. I 

 discovered what to me is a new and interesting sequence 

 in the numbers which constitute the answers to squares of 

 successively and uuiformly increased dimensions. 



I think the matter will interest some of your readers. 



