fifth decimal place. The sum of the six terms is 1.34392, and the accumulated value of the loan is 740 x 1.34392 pounds, or £994 10s. Od. In this case it is doubtful whether the use of the binomial series saves any time over direct calculation. Presently we shall have calculations for which there is no comparison between the methods. EXERCISES. 2. A railway has 34 stations. How many kinds of third class single tickets does it take to supply the system? 3. A school has 87 girls, and each of them sends a Christmas card to every other. How many do they send? length W b r 4. The floor of a conservatory is to be paved with red, white, and black tiles according to the design indicated in Fig. 2. If the length and breadth of the floor are respectively 40 and 20 times the length of the diagonal of a square tile, find the number of square tiles of each colour that will be required, ignoring the smaller tiles. 5. In the game of dominoes a number of blocks are used. Each block is marked off into two parts, and on each of these parts a number of pips, between O and 6 inclusive, are marked. These are arranged in all possible ways, and the pips on the breadth ་ 2 b 22 20 W FIG. 2. a 772 two parts of a block may be the same or different; and no two blocks are alike. How many blocks are there? 6. A publisher uses sheets of paper 24 inches by 10 inches to make a book each page of which measures 8 inches by 5 inches. He folds each sheet of paper so that it makes 6 leaves or 12 pages. Sketch a sheet of paper, back and front, mark the creases and write on the several pages the numbers 1, 2, . . . 12, so that when the sheet forms part of the book these numbers will appear in proper order right side up. Mark corresponding corners of the back and front views of the sheet with the letters A, B, C, D. 7. Nine cards numbered 1 to 9 are arranged in three heaps as in Fig. 3. The bottom card of heap 1 is taken away and placed on top of heap 2; then the card now at the bottom of heap 1 is taken and put on heap 3. Similarly, the bottom card of heap 2 is placed on top of heap 3; and the card left at the bottom of heap 2 put on heap 1. Similarly, the card at the bottom of heap 3 is placed on heap 1, and the card so left at the bottom on heap 2. Sketch the cards in their position at this stage. When this operation is repeated, name the positions taken up in succession by card No. 1. After how many repetitions does this card return to its original position? Are the other cards then in their original positions? How do you know? 8. A marble statue is 5 feet 6 inches high. A statuette which is an exact copy of the statue on a smaller scale and in plaster weighs lb. The statuette is 9 inches high and is half the weight it would have been in marble. What is the weight of the statue, to the nearest hundredweight? Note that : 9. A wooden ball 25 centimetres in diameter weighs 5.9 kilograms. Find, to the nearest hundredth of a gram, the weight of a cubic centimetre of the wood. The volume of a ball of diameter d is d3, where π = 3.14. 10. The number of cubic feet of timber in an oak log, which tapers uniformly, may be calculated from the formula 0.2618h × (a2+ab+b2), where a and b are the numbers of feet in the diameters of the ends of the log, and h is the number of feet in the length of the log. Find what is the weight of such a log of oak whose length is 15 feet, and the diameters of whose ends are 3 feet and 2 feet respectively, assuming that a cubic foot of the oak weighs 50 lb. Give your answer to the nearest cwt. If you assumed the diameter to be 2 feet 6 inches throughout, how much per cent. (to the nearest integer) would the result be wrong? 11. Find the value of 1 + mn + m (m-1) n2 + m (m-1) (m-2) n3 (1) when m = 2 and n = 3, and also (2) when m = 24 and n = 1 • 12. The volume and surface of a sphere are respectively π+3 and 42, r being its radius. A sphere S, whose radius is 10 centimetres long, is covered with a substance which is everywhere 1 millimetre thick. Show that the volume of the substance is nearly equal to the surface of S multiplied by the thickness of the substance, and find the percentage error involved in assuming this to be the case. 13. The present value of an annuity A for r per cent. compound interest is equal to n years reckoned at A, where Find the present value of an annuity of £50 for 15 years at 24 per cent. 14. In Fig. 4 DC is the graph of y Express the area of OBCD in terms of x, y=ax+b = ax + b, and OB = x1. a and b. If such a line as DC is drawn, and it is found that for all values of x the area of OBCD is рx12 + cx1, what must be the equation of DC? 15. By drawing find the sines of 23° and 74°. Make (when possible) angles whose sines are 0·21, 0·42, 0·84, 1.68, and measure them with your protractor. Check your results by referring to a book of tables. 16. Draw two straight lines AOB and COD crossing at right angles. On tracing-paper draw a line PQ 8 cm. long, and on PQ mark R 1 cm. from P. Move the tracing-paper so that P always lies on AB and Q on CD. Prick through the point R in various positions, and so draw the path of R. Repeat, taking R in succession at distances of 2, 4, 6, 7 cm. from P. In the case when R is midway between P and Q, discuss by general reasoning the nature of the path of R. 17. Draw a straight line and a curve between two points A and B. Erect perpendiculars at intervals of 1 centimetre along the straight line AB. Join the points at which successive perpendiculars meet the curve. Estimate approximately the area of the figure by assuming it to be equal to the polygon so made. 18. Water runs at 4 miles an hour along a pipe the cross-section of which is 3 square feet. How many gallons (to the nearest million) does it deliver in 24 hours? A cubic foot is 6.23 gallons. 19. On tracing-paper draw lines OX and OY, meeting at an angle of 30°, making OX 10 cm. and OY 7.5 cm. long; joining these lines draw others X1 Y1 X2Y2 XY3. parallel to XY and 0.4 cm. apart. Draw a semi-circle bounded by a diameter, and mark T the centre and S any point within the boundary. Place the tracing-paper on this semi-circle with O above the point S, and pin it at this point. Now rotate the tracing, and as any point on OX (say X1) comes over the boundary of the semi-circle, prick through Y the corresponding point on OY; when X2 is on the boundary prick through Y2; and so on. the complete locus of Y as X traces out the given figure. What would the locus of Y be if O were pivoted at T? Draw CHAPTER XI ON CARRYING WATER 1. A MAN has to take a bucket from a cottage C, fill it at some point P of a ditch D, and carry it to a trough T. Taking the distances as shown in Fig. 1, find, by drawing to a suitable scale, the length of the journey for different positions of P, say at distances of 5, 10, 15, . . . metres from A; and show your results in a table. At every point P draw a 1 to the ditch, of a length that represents (on the scale chosen) the length of the journey for that position of P. By drawing a curve through the ends of these Ls, make a graph showing the length of the journey for any position of P. From your graph find the position of P that gives the shortest journey. 2. Produce TB to U so that BU = BT, that is, take U the image of T in the line of the ditch. Suppose the man to go from C to U, crossing the ditch at some point P, |