From these it is inferred, and by mathematical induction it may be proved that = = (Tna+Tn-1)2 = ⚫). T2(a+1)+2aTaTn-1+(Tn−1)2 [(T2)2+2T2T2-1]a +[(Tn)2+(Tn−1)2]. Note that a[= (1+√5)/2] is irrational and, consequently, if l, m, p, q, are integers the statement that implies l = la +m = pa + q p and m = q. For, if this were not the case, we would have the irrational a equal to the rational quantity, (g-m)/(l-p). Hence, T2n = (Ta)2+(Tn−1)2, T2n+1 When n-1 is put for n in the latter, it can be changed into the second of the two desired relations. The same method can be used should we wish to prove III. (Tn)3+(Tn_ı)3−(Tn−z)3. By the Editor. It is curious to see into what realms of mathematics we are frequently led by some problem. The above series, for example, arose from a problem which the Proposer had cut from a newspaper. A discussion of the nth term of the series is found in the American Mathematical Monthly, vol. 28, page 329. Also solved by F. Howard. 717. Proposed by Daniel Kreth, Wellman, Iowa. Construct the triangle, given A, the length d of the bisector of this angle, and the sum e of the including sides AB and AC. = Id, the bisector. Solution by H. R. Scheufler, Culver Military Academy, Culver, Ind. Construct the given ZA, bisect it, and make AE From E draw lines parallel to the sides of the given angle, forming the rhombus AFED. From e, which is the sum of AB and AC, subtract 2AD and then divide the remainder into two parts, ƒ and g, such that AD will be a mean proportional between the two parts. Then AD +f is AB, one side of the desired triangle; and AF+g is AC, the other side. The construction will be proved correct if we prove that BEC is a straight line, which can be done by proving As BDE and EFC similar. By construction_BD/AD AD/FC. But AD DE EF, so that BD/DE EF/FC. Also ZBDE ZEFC, and the triangles are Analytic solutions using trigonometry showing how AB and AC could be found by solving a quadratic were received from Moe Buchman, student, College of the City of New York; Henry L. Wood, Boonton, N. J.; and the Proposer. The following trigonometric solution, however, can easily be carried out. II. Solution by Michael Goldberg. == The area of AABC equals the product of e/2 and h, where h is the perpendicular from the end of d to either side. The area is also equal to 1/2AB XAC sin A. Hence he esc A = ABXAC. Accordingly, we first find x, the mean proportional between h and e csc A. Then we divide the line e into two parts for which x will be the mean proportion. The two parts will be the desired lines AB and AC. 718. Proposed by F. Howard, San Antonio, Texas. A clock has three hands, the hour, minute and the second, on the same pivot. What is the first time after 12 o clock when the hands will be equally distant? I. Answer by Elmer Schuyler. According to the following solution, given in Ghersi's Mathematica Dilettevole e Curiosa, it is impossible. Let the hour hand be a minute spaces beyond XII, then the minute hand will be x +20 beyond XII, and the second hand be x+40; or, else the second hand is x+20, and the minute hand is x +40. In the first case, we must have where n and n1 are integers. Eliminating x, gives (3n+1)/(3n1+2) = 11/719 or 3(11n1-719n) But 11n1-719n is an integer; then 697 must be divisible by 3 in order that the problem be possible. The second case gives 3(11n1-719n) for the same reason. = 1,427 which is again impossible Similarly explained by Michael Goldberg, and the Proposer. Assuming that the problem meant that one hand should be midway between the other two, solutions were received from T. E. N. Eaton, Redlands, Cal.; Arthur H. Lord, Lynn, Mass.; Edward Lewis, 23, Redlands H. S., Calif.; Daniel Kreth, Wellman, Iowa; Eva Tilton, 23, Redlands H. S.; A. M. Waas, Philadelphia, Pa. After 59 13/73 seconds the hour hand is midway between the other two; and after one and 13/1427 minutes the second hand is between the other two. 719. Proposed by Walter R. Warne, Pennsylvania State College, State = 435, CA A, B, C are three buoys; AB 320, BC 600. A ship S finds that AB subtends an angle of 8°, and BC an angle of 26°. How far is the ship from each of the buoys? There are four possible solutions to this problem. Michael Goldberg was the only one to find the numerical values for all four cases; and Daniel Kreth showed the solutions for the general case when the sides are a, b, c, etc. Other solutions were by Moe Buchman (2 cases), F. Howard (2 cases), and Karl Paul, Waubun School, Minn. This problem deserves more space than can be given here, and so the editor has turned over all the solutions to Daniel Kreth with the suggestion that he write a special article for us about this problem. We hope it will appear shortly. 720. For high school students. Proposed by the Editor. A can do a certain piece of work in 25 days, B in 22, C in 20 days. A starts on the job and after working 3 days, hires B to assist him; then three days later C also begins working. How soon is the job completed? Solution by Lois A. Woodbury, Nashua H. S., New Hampshire. Since the total amount of work done is equal to the whole piece of work, (6+x)/25+(3+x)/22+x/20 : 1; I = 4.60+ = Then, since A worked all the time, the time necessary to do the work was 6+x 10.60+ days. Too many solutions have been received to be able to mention all the names. The best ones were: Henrietta Humbert, Jerome, Ariz.; Ysabel Hastings and Austen Roach, Redlands, Calif.; Violet Lord, and Fred W. Peaslee, Nashua, N. H.; and for solutions by arithmetic, Olin W. Munger, Northeast H. S., Kansas City, Mo.; and Eric McCann, Grant's Pass, Oregon. The chief faults of the poor solutions were that they contained such expressions as Let x when C begins, or r time to finish or the equation was stated with no explanation of how it was derived. = = PROBLEMS FOR SOLUTION. Solutions should reach the editor by the twentieth of the month following publication. 731. Proposed by F. Howard, San Antonio, Texas. A geometric and an arithmetic progression have the same pth, qth, and rth terms, a, b, and c, respectively. Prove a(b-c)loga+b(ca)logb+c(a - b)logc = 0. 732. Proposed by Elmer Schuyler, Bay Ridge H. S., Brooklyn, N. Y. BCA is a diameter, CA CB = = 1. Draw the semicircle AGB, G being its midpoint. With A as a center, and AG for a radius, draw an arc cutting AB in D. With A as a center, and AC as a radius, draw an arc cutting the semicircle in M. Show that DM is approximately 2. (Peraux's Approximation.) 3 733. Proposed by Harris F. MacNeish, College of the City of New York. Find without using trigonometry the volume of a regular dodecahedron in terms of the edge e. 734. Proposed by Norman Anning, Ann Arbor, Mich. 0, show that a2b+b2c+c2a 735. For high school students. Proposed by the Editor. Prove: the two common external tangents of two circles intercept on a common internal tangent a segment, CD, equal to the external tangent AB. The examination below was received from John Lundberg, Goteborg, Sweden. To judge the nature of any work from an examination we need to know something about the age of the pupils, etc., and the following information has been furnished by C. R. Nilsson, Cleveland, Ohio, who is a graduate of the Goteborg schools and a forme. student at the University of Illinois. The pupils enter at an average age of ten years. The work is divided into seven classes, a lower school of five classes and an upper school or "gymnasiet" of four classes called the lower and upper sixth and the lower and upper seventh. Thus the work covers nine years, and the pupil takes his graduating "Studentexamen" when he is nineteen. After finishing the lower school, the pupil must choose between the "Real gymnasiet" which prepares for business and technical schools, and the "Latin gymnasiet" which prepares for the professions of law, medicine or ministry. The examination below is for those ready to graduate from the Latin gymna siet. 1. In a town, 125,250 different connections can be made over the telephone system. How many subscribers are there? 2. Separate 365 into two parts such that the sum of the square roots of the parts will be 27. 3. Three numbers whose sum is 31 form a geometric progression. If the second term is increased by 8, the numbers will form an arithmetic progression. Find the numbers. 4. A stream of light in the shape of a cone lights up one-third of the surface of a sphere whose radius is r. How far from the surface of the sphere is the vertex of the cone? 5. Express the cube root of the repeating decimal 4.629629, as a repeating decimal. 9.46 is parallel to CD 6. In the isosceles trapezoid ABCD, AB 7.13. Angle ABC = 72°. Find the lengths of the diagonals. = 7. The logarithm of a number to the base 10 equals the sum of its logarithms to two other bases, of which one base is one-tenth of the other. What are the two bases? 8. For pail shaped like the frustum of a cone, the areas of the bases are 15 dm.2 and 10 dm.2 and the height is 12 dm. The pail contains water at a temperature of 4°. What is the pressure on the bottom? SCIENCE QUESTIONS. Conducted by Franklin T. Jones. The Warner & Swasey Company, Cleveland, Ohio. Readers are invited to propose questions for solution-scientific or pedagogical-and to answer questions proposed by others or by themselves. Kindly address all communications to Franklin T. Jones, 10109 Wilbur Ave., S. E., Cleveland, Ohio. Please send examination papers on any subject or from any source to the Editor of this department. He will reciprocate by sending you such collections of questions as may interest you and be at his disposal. Foreign Examination Papers. The Editor of this department is not discouraged by failure to obtain responses to his request for foreign examination papers. Did not Mr. John Lundberg of Goteborg send some examinations from Sweden? Speaking mathematically the chance of obtaining the papers desired from France, Italy, Spain, Belgium, Norway, Denmark and other countries is increasing the event is bound to happen some time, hence each month that passes without a favorable reply increases the Editor's chance of obtaining the papers the next month. Once more, readers of SCHOOL SCIENCE AND MATHEMATICS in the countries mentioned above, please send examination papers to the Editor or inform him how they may be obtained through book sellers or education departments. Do the universities of India give entrance examinations? 389. Problems for Solution. From an advertisement in The Country Gentleman. One pulls big stumps (1) What is the mechanical advantage of the PULLEY system in this stump puller? (2) Adopt values for length of lever and toggle where fixed end of lever is attached, also for push of a 180-pound man and figure the force produced at the stump? 390. Submitted through Mr. Charles M. Turton. A vessel contains 11.83 cu. ft. of air at a pressure of 33.3 lb. per sq. in. It is desired to increase the pressure to 40 lb. per sq. in. by supplying air from a second vessel which contains 19.6 cu. ft. of air at a pressure of 60 lb. per sq. in. What will be the pressure in the second vessel after the first has been increased to 40 lb. per sq. in.? Examination Papers. The following paper in Physical Science completes the set of science papers from the Province of Alberta whose publication was undertaken in the course of the past year or more. This set of papers deserves careful study, representing as it does the course of study in science in Alberta schools. PROVINCE OF ALBERTA. HIGH SCHOOL AND UNIVERSITY MATRICULATION EXAMINATIONS BOARD, DEPARTMENTAL EXAMINATIONS, 1920. GRADE XII.-PHYSICAL SCIENCE. Time-Two and one-half hours. (a) Define the terms: dyne, erg, kilowatt, horsepower. (b) Compare the kinetic energy of a ton truck moving at the rate of 15 miles an hour with that of an automobile of 1,500 pounds weight moving at the rate of 20 miles an hour. (c) Show, using a diagram, how a system of pulleys may be arranged so that, neglecting friction, a body weighing 900 pounds may be raised by applying a force of 150 pounds. (a) By reference to the relation between pressure and depth in a liquid, demonstrate how great must be the buoyant force of a liquid on a body immersed in it. (b) What is the rate of pressure measured in kilograms per square dm. at a depth of 4 m. in a pond if the barometer stands at 75 cm. and the specific gravity of mercury is 13.6? Write a note on the molecular theory showing how it helps to explain the following: (a) The unequal rates of diffusion of gases through a porous partition. (b) The expansion of a solid when heated. (c) The magnetization of a bar of iron. (a) (1) Describe the movements of air particles when transmitting sound. (b) (c) (2) Explain the presence of a node in the vibrating air column of a tube open at both ends. Describe an experiment to illustrate the conditions under which beats may be produced. Explain this phenomenon. Show by reference to musical instruments: (1) How the laws of transverse vibrations of strings are applied in producing notes of different pitch. (2) How the intensity of the sounds produced is increased by consonance. (a) Describe an experiment to show how the heat of vapourization of water may be determined. (b) Show by reference to the turbine engine how the energy |