410 



THE POPULAR EDUCATOR. 



apparatus being fixed vertically in front of an observer, he can 

 perceive by the reflection produced by the silvered sectors his 

 own image and generally those of other objects and persons 

 situated in front of the star. He can also see through the open 

 spaces persons and other objects placed behind. No optical 

 illusion can result from this super-position of two visual 

 phenomena, because the observer distinguishes the apparatus 

 and becomes acquainted with its action. But supposing that 

 this star takes, round an horizontal central axis, a very rapid 

 movement of rotation, the reflecting sectors are no longer dis- 

 tinguishable from the vacant spaces. The observer will think 

 that he sees only a plate of glass, almost invisible, like window- 

 glass, and he will behold, not without astonishment, the images 

 of the anterior objects superposed more or less confusedly on 

 the objects behind ; strange illusions must be the result. 



Similar phenomena are produced by means of an unsilvered 

 sheet of glass, when the images before the reflecting surface are 

 strongly illuminated by the oxy-hydrogen light, whilst the real 

 figures behind the glass are more moderately lighted by gas jets. 

 The spectral figures seen behind the unsilvered glass are of a 

 peculiar paleness, and yet their resemblance to the real figures 

 is so great that many spectators cannot tell which is the ghost 

 or which the real figure walking on the stage. 



The transparent mirror described above will serve the same 

 purposes as unsilvered glass, without requiring the same mode 

 of lighting. M. Lucas constructed a small apparatus in which 

 this arrangement was successfully carried out. There are four 

 radiating sectors, and the means by which they are fixed and 

 traversed by a geared shaft are hidden under tho table. A slit 

 parallel to the foremost edge gives passage for tho metallic 

 blades. By simple gearing the operator can obtain, by turning 

 a handle, a rotation sufficiently rapid to produce the effect of an 

 unsilvered vertical or semi-circular sheet of glass. The edge of 

 this silvered glass apparatus is concealed by a wooden frame. 



A person being placed behind tho rotating mirror at a 

 distance of six and a half feet from tho vertical plane passing 

 through the axis of rotation, another is placed in front of the 

 mirror at the same distance of six and a half feet from the 

 other side of the vertical plane, when the latter sees two persona 

 side by side, one being tho spectral reflection produced by the 

 revolving radiating sector of silvered glass, and the other the 

 real person. Tho mirror can be made of metal silvered and 

 polished on both sides ; the effect obtained is reciprocal, and 

 tho illusion is the same for the two persons. 



Another effect was obtained by M. Lucas by placing a 

 common mirror vertically at the distance of two and a half feet 

 behind an unsilvered mirror. The reflected image is repeated a 

 number of times at greater and greater distances, as is the 

 case with an object placed between two mirrors. Yet while in 

 the latter case the successive images are in reversed positions, 

 those in the apparatus are face to face. 



Fig. 5 represents the beautiful effect produced in any room 

 by placing two looking-glasses opposite to each other one 

 over the chimney-piece, the other on the wall facing it ; thus a 

 chandelier in the centre is reflected over and over again, whilst 

 the room being also multiplied gives the spectator the idea of 

 endless apartments, all leading one into the other. At one of 

 the theatres, where the manager was anxious to represent a 

 magnificent room hung with mirrors, real ones were used, and 

 it was only by the laughter of the audience that he discovered 

 he had not taken sufficient precaution to exclude the reflection 

 of the carpenters and scene-shifters, who were all shown up in 

 the elegant apartment with tho grandly-dressed " prima 

 donna." Thus, upon the stage, a painted scene representing a 

 mirror is more convenient than the real silvered glass, although, 

 when laid at a slight angle on the stage, looking-glass has 

 been used with remarkable success for the purpose of imitating 

 the still water of a beautiful lake. 



LESSONS IN ALGEBRA. XXV. 



EXERCISE 41. ALGEBRAICAL PROBLEMS (continued). 

 33. A, B, and C possess certain sums of money, such that, 

 if A receive, in addition, half what B and C have, he will possess 

 a ; if B receive, in addition, a third of what A and C have, he 

 will possess .6 ; and if G receive, in addition, a fourth of what 

 A and B have, he will possess c. What sum has each ? 



34. All that is known respecting the co-efficients a, b, and 

 the quantity c, in the trinomial expression ax- + bx -f- c, is, that 

 when 4 is put for x, the value of the expression is 42 ; that when 

 3 is put for x, its value is 22 ; and that when 2 is put for x, its 

 value is 8. What are the values of the co-efficients a, b, and 

 the number c ? 



35. Five persons engage in play on the condition that he who 

 loses shall give to each of the others as much as he already has. 

 All lose in their turn, and yet at the end of the fifth game they 

 all have the same sum, namely, ,32. How much did each 

 begin with ? 



36. Solve the general simultaneous equations to two unknown 

 quantities, denoting the co- efficients of x and y in the first 

 equation by a and 6, and the absolute term by c and the co- 

 efficients of x and y in the second equation by d and c, and the 

 absolute term by /; and let all the terms in both equations bo 

 positive. 



37. A number consists of three digits, of which the difference 

 between the first and second is the 'same as the difference 

 between the second and third. If the number be divided by tho 

 sums of the digits, tho quotient will bo 26 ; but if 198 be added 

 to it, the digits will be inverted. Eequired the number. 



38. A person distributes a shillings among n persons, men 

 and women : to the men ho gives p pence each ; to the women 

 q pence each. Eequired the number of men and the number of 

 women. 



39. If from a vessel of wine containing a gallons, b gallons be 

 drawn off, and the vessel then filled up with water, and this 

 operation be repeated n times successively, find the quantity 

 of wine remaining after these n operations are finished. 



40. In a garrison of 2,744 men, there are two cavalry soldiers 

 to twenty-five infantry, and half as many artillery as cavalry. 

 Find the numbers of each. 



41. A person dies worth ,13,000 ; some of this he leaves to 

 a charity, and twelve times as much to his eldest eon, whose 

 share is half as much again as that of each of his two brothers, 

 and two-thirds as much again as that of each of his five sisters. 

 Find the amount of tho bequest to charity. 



42. A farm of 270 acres is divided among A, B, and C .- A has 

 7 acres to 11 acres of B, and C has half as much again as A and 

 B together. Find the shares. 



43. The length of a floor exceeds the breadth by 4 feet; if each 

 had been increased by a foot, the area of the room would have 

 been increased by 27 square feet. Find its original dimensions. 



44. A sum of money was left for the poor widows of a parish, 

 and it was found that if each received 4s. 6d. there would be Is. 

 over; whereas, if each received 5s., there would be 10s. short, 

 liow many widows were there, and what was the sum left ? 



45. There is a number consisting of two digits whoso differ- 

 ence ia two ; and if it be diminished by half as much again as 

 the sum of the digits, the digits will be inverted. Find it. 



4G. A person has travelled altogether 3,036 miles, of which 

 he has gone seven miles by water to four miles on foot, and five 

 miles by water to two on horseback. How many did he travel 

 each way ? 



47. A cistern can be filled in 15 minutes by two pipes, A and 

 B, running together : after A has been running by itself for 5 

 minutes, B is also turned on, and the cistern is filled in 13 

 minutes more. In what time would it be filled by each pipe 

 separately ? 



48. A mass of copper and tin weighs 80 pounds, and for 

 every 7 pounds of copper there are 3 pounds of tin. How much 

 copper must be added to the mass that for every 11 pounds of 

 copper there may be 4 pounds of tin ? 



49. A does jj of a piece of work in 10 days, when B comes to 

 help him, and they take 3 days more to finish it. In what timo 

 would they have done the whole, eauh separately, or both to- 

 gether ? 



50. Divide .149 among A, B, C, and D, so that A may have 

 half as much again as B, and a third as much again as B and 

 C together ; and D a fourth as much again as A and C to- 

 gether. 



51. There are two silver cups, and one cover for both. The 

 first weighs 12 ounces, and, with the cover, weighs twice as much 

 as the other cup without it; but the second with the cover 

 weighs a third as much again as the first without it. Find the 

 weight of the cover. 



52. A man could reap a field by himself in 20 hours, but with 



