Jdne 30, 1882.] 



• KNOWLEDGE 



81 



by doubling the odd number and subtracting 1 oi- 15, and doubling 

 the even number and substracting 2 or 16. With 9 join numbers 

 formed in the same way from those with 5. This will complete 

 four groups, and the fifth group will consist of the remaining 

 numbers, either 4, 6, 10, or 8, 12, 14. 



Make the second day's arrangement by adding 2 to every number 

 in the first day's, except 13 and 14, from which you must subtraci 

 12. Make each day's arrangement from that of the proceding day 

 in the same way, and the problem is solved. 



Any solution may be varied by making the odd and even numbers 

 change places — that is, each odd number changing places with the 

 next greater even number. 



The explanation of this solution is that in the first day's arrange- 

 ment one odd number is combined with another differing from it by 

 2, one with another differing from it by 4 or 10, and one with 

 another differing from it by 6 or 8 ; and it is the same with the 

 even nimabcrs. Also the combinations of odd and even numbers 

 are such that one odd number is combined with an oven number 

 1 greater, one with an even number 3 greater, 



one with an even number 5 greater or 9 less, 

 7 .. 7 „ 



9 „ 5 „ 

 11 .. 3 „ 

 I. » J) 1 less. 



These differences are not changed by the successive additions of 2, 

 except by changing a difference to what we may call its comple- 

 ment, as + d to -f (J — 14. The place of every number in the first 

 day's arrangement will in the coui-se of the six following days bo 

 occupied by every other number of the same class (odd or even as 

 the case may be), and consequently every number will, in the 

 course of the seven days be once joined with C, and once with 

 a number differing from it by each possible difference, whether 

 the difference be added or subtracted. 



This method is applicable with slight modifications to most other 

 Bcboolgirl problems, where the number of the girls is 13 n + 3, and 

 not otherwise ; but in some cases, and as 1 have particularly 

 noticed in cases of 60 ?i — 9 girls, there are difficulties in the way 

 of applying it which 1 do not know aay simple way of getting over. 

 For better illustration of the method of solving other problems, 1 

 add one solution of the problem of thirty-nine schoolgirls. 



Make a series of odd numbers beginning with 3, by doubling each 

 term and subtracting 1 or 39 to form the next, and set the series 

 down in pairs thus : — 



3 5 8 



9 17 26 With each of these pairs join the 



33 27 22 oven number which is the sum of 



15 29 6 the two numbers composing the 

 19 37 18 pair, deducting 38 if greater than 

 35 31 28 38. These numbers are the same 



23 7 30 as the even terms in the series of 

 13 25 38 odd numbers, each increased by 1. 

 11 21 32 Now take the odd terms in that 



— series in order, add 1 to each, and 



4 10 34 set them down in threes, as is here 



16 20 36 done. 



24 14 12 



Those sets of three, with the addition of C, 1, 2, form the arrange- 

 ment for the first day, from which the arrangements for the other 

 days are formed by successive additions of 2 or subtractions of 36. 

 Algkrnon Urav. 



MISPLACED SPA. 



[451] — The efflorescence mentioned by "Curiosity" has been 

 found by chemical analysis to vary considerably in composition. It 

 often consists of sulphate of magnesia, also sulphate of lime, of 

 carbonate sulphate, or nitrate of soda, of chloride of soda and potash, 

 and carbonate of potash. It is attributable sometimes to the bricks 

 or stones of a wall, sometimes to the mortar. 



In bricks burnt with coal fires or made from clay containing iron 

 pyrites the sulphur from the fuel converts tho lirao or magnesia in 

 the clay into sulphates. When tho bricks are wet thcso dissolve ; 

 wlion dry they evaporate, leaving crystals on the surface. Many 

 limestones oonuiin magnesia; these are acted upon during calcina- 

 tion by the sulphur in tho fuel ; sulphates nro formed which find 

 their way into tho mortar and produce tho oilioroscenco mentioned. 

 Again, the sulphur acids evolved from ordinary house fires attack 

 tho magnesia and lime in tho mortar joints of chimneys ; these dis- 

 solve and evaporate on the surface. Tho formation of chlorides is 

 nearly sure to take place if sea sand or sea water be used, or in 

 bricks made from cl.ay which has been covered by salt water. 



" Prevention is better than cure." 



The best plan is to avoid all tho materials above mentioned as 

 likely to give rise to efflorescence, in tho case of bricks; clay con- 



taining pyrites or much magnesia should not be used ; special bricks 

 may bo burnt with coke or wood. 



As regards mortar, tho use of limestones containing magnesia to 

 any gieat extent may generally be avoided. If, however, it does 

 occur, in spite of all precautions, the following remedies may be 

 tried : — 



In the case of ashlar work, the surface may 'bo covered with a 

 wash of powdered stone, sand, and water, which is afterwards 

 cleaned off. Tliis fills up the pores of the stone, and temporarily 

 stops the effiorescence. When tho wash is removed the efHorescenco 

 will recommence, but in a weaker degree than before. 



Painting tho surface is sometimes eflicacions if it is done before 

 the efflorescence commences. 



The mortar before use may be treated, to prevent it from causing 

 efflorescence by mixing with it any animal matter. 



W. H. Peetwee. 



EXTINGUISHING CANDLE-FLAME. 



[452] — I do not know whether it has ever come under your notice 

 that an ordinary wax candle may be blown out without leaving any 

 red-hot snuff. Jf the candle be drawn backwards so that tho flame 

 envelopes the wick right to the end, the burning of the latter is 

 quenched, for the obvious reason that it is in the heart of the flame 

 where no air can reach it, and a sudden puff at this juncture puts 

 out the flame and wick too. Tho proper direction to blow in is that 

 of the tangent to the curve of the wick from the end of tho latter j 

 in fact, the experiment can, with care, be made to succeed withoat 

 moving the candle at all, if attention be paid to this point. 



I have also noticed that tho third envelope of a candle-flame can 

 be made distinctly visible by burning a little sodium in the neigh- 

 bourhood of tho flame, so that, from being faintly blue, it becomes 

 tinged with yellow, and can be seen, in spite of the luminous layer 

 inside it. C. L. B.vbxes. 



SECTIONS OF A CONE. 

 [453] — In drinking a glass of water, as the glass is sloped 

 towards the lips the surface assumes an elliptical form. It has 

 strnck me that since it was thus possible to obtain a section of a 

 cylinder, a somewhat similar contrivance might be used for demon- 

 strating the figures commonly known as " sections of a cone." My 

 idea is simply this — to make a hollow glass cone, and partially fill 

 it with any tinted liquid. By giving it different degrees of inclina- 

 tion, the surface of the liquid might be made to assume the form of 

 any conic section, from the circle to the hyperbola. Further, it the 

 apparatus were so arranged that the liquid occupied exactly half 

 the capacity of the hollow cone, we conld obtain tho limiting form 

 of two straight lines, by holding the cone so that its axis should bo 

 horizontal. I think tho advantages of this system are many and 

 obvious. First, it would bo cheaper than the wooden sections now 

 in use, which requii'e delicate workmanship ; secondly, it would 

 have the advantages of comjiactness and completeness, and would 

 run no danger of being lost piecemeal ; and, lastly, the demon- 

 strator, by a motion of tho hand, could actually show the ellipse in 

 the process of passing off into cither of its two limiting forms — 

 namely, tho circle or the parabola — and thus attract tho attention 

 and interest of the pupil. George E. Cbawkoru. 



Amongst tho coming electrical exhibitions, that to bo held at 

 Munich in the autumn will be of csjiocial interest, ns it is tho 

 intention of tho authorities to use, as tho driving-power for tho 

 dynamo-machines, tho water-force of the river lear, estimated at 

 3,000 horso-power, and to prove that it is possible to make use of 

 it to light, at a distance of some miles tho streets and houses, and 

 to distribute tho force to the different workshops. 



Pricks or Books. — Tho following very sensible paragraph 

 appears among tho answers to correspondents, in our valuable con- 

 tempornry, tho Schnul llnaid Chronicle : — " ' Educationist ' writes : 

 '1 should' bo much obliged if yon would add tho price of the books 

 you so excellently rovicw,' observing that 'thoy may bo 8ix|>oncc 

 or a guinea, and no ono can toll.' — Wo have had from time to lime 

 many commnnications from our readers in various jmrts of the 

 country to the game effect. We can only say that it would give us 

 great pleasure to affoni this information to our reailors^ but wo owe 

 also some consideration to tho proprietors of tho School /i.xinJ 

 Chronicle, who naturally consider it to bo tho duly of thi- publii>hcrs 

 of the books to inform tho public through our advortising columns 

 of tho jiriccs of the books which wo review. In a largv pro|K)rtion 

 of cases this information will bo found auioiig our ndverlisonionts, 

 but not nccossorily in tho same week in which tUo review appoara- 

 We think publishers would consult their own interest by announcing 

 tho prices of tho books inimcdiatoly after tho api>carauco of tho 

 reviews." 



