Q A^ UC-NRLF s/? H3 s *B S^3 474 roDUCi TO GRAPHICAI A L Cr E B R A H*S. HALL )r^ S7 IN MEMORIAM FLORIAN CAJORl ^^'^ (2^^ V^'^'X,.-''^.,^ ^-"i^««««»«g,6^ ^—- ^u x=4, y = 4x(-7)--28. and so on. GRAPHICAL ALGEBRA. l?/*proceeding iri ' this way we can find as many values of the function 3^8 w^r please.-. But We are often not so much concerned ■^wiih ,fciie^'4cti:|Vii y|il,u^s,' which a function assumes for different 'vahies 'of the varial3le as with the icay in which the 'value of the function changes. These variations can be very conveniently represented by a graphical method which we shall now explain. 4. Two straight lines XOX\ YOY' are taken intersecting at right angles in 0, thus dividing the plane of the paper into four spaces XOY^ YOX\ XOY\ Y'OX, which are known as the first, second, third, and fourth quadrants respectively. Q Fig. I. The lines XOX, YOY' are usually drawn horizontally and vertically ; they are taken as lines of reference and are known as the axis of x and y respectively. The point is called the origin. Values of a.^ are measured from along the axis of ^, according to some convenient scale of measurement, and are called abscissae, positive values being drawn to the right of along OX, and negative values to the left of along OX'. Values of y are drawn (on the same scale) parallel to the axis of y, from the ends of the corresponding abscissae, and are called ordinates. These are positive when drawn above X'X, negative when drawn helow X'X. 5. The abscissa and ordinate of a point taken together are known as its coordinates. A point whose coordinates are X and y is briefly spoken of as " the point (^, ?/)." The coordinates of a point completely determine its position in the plane. Thus if we wish to mark the point (2, 3), we PLOTTING A POINT. 6 take e^• = 2 units measured to the right of 0,7/ = 3 units measured perpendicular to the .r-axis and above it. The resulting point P is in the first quadrant. The point ( - 3, 2) is found by taking ./' = 3 units to the left of 0, and ?/ = 2 units above the ^-axis. The resulting point Q is in the second quadrant. Similarly the points ( - 3, — 4), (5, - 5) are represented by R and S in Fig. 1, in the third and fourth quadrants respectively. This process of marking the position of a point in reference to the coordinate axes is known as plotting the point. 6. In practice it is convenient to use squared paper ; that is, paper ruled into small squares by two sets of equi- distant parallel straight lines, the one set being horizontal and the other vertical. After selecting two of the intersecting lines as axes (and slightly thickening them to aid the eye) one or more of the divisions may be chosen as our unit, and points may be readily plotted when their coordinates are known. Conversely, if the position of a point in any of the quadrants is marked, its coordinates can be measured by the divisions on the paper. In the following pages we have used paper ruled to tenths of an inch, but a larger scale will sometimes be more convenient. See Art. 26. Example. Plot the points (5, 2), ( - 3, 2), ( - 3, - 4), (5, - 4) on squared paper. Find the area of the figure determined by these points, assuming the divisions on the paper to be tenths of an inch. Taking the points in the order given, it is easily seen that they are repre- sented by P, Q, JR, S in Fig. 2, and that they form a rectangle which contains 48 squares. Each of these is one-hundredth part of a square inch. Thus the area of the rectangle is '48 of a square inch. FiJ.5: GRAPHICAL ALGEBRA, EXAMPLES I. [The following examples are intended to he done mainly by actual measurement on squared paper ; where possible^ they should also be verified by calculation.'] Plot the following pairs of points and draw the line which joins them : 1. (3, 0), (0, 6). 2. (-2,0), (0, -8). 3. (3, -8), (-2,6). 4. (5,5), (-2, -2). 5. (-2,6), (1, -3). 6. (4,5), (-1,5). 7. Plot the points (3, 3), ( - 3, 3), (-3, -3), (3, -3), and find the number of squares contained by the figure determined by these points. 8. Plot the points (4, 0), (0, 4), ( -4, 0), (0, -4), and find the number of units of area in the resulting figure. 9. Plot the points (0, 0), (0, 10), (5, 5), and find the number of units of area in the triangle. 10. Shew that the triangle whose vertices are (0, 0), (0, 6), (4, 3) contains 12 units of area. Shew also that the points (0, 0), (0, 6), (4, 8) determine a triangle of the same area. 11. Plot the points (5, 6), (-5, 6), (5, -6), (-5, -6). If one millimetre is taken as unit, find the area of the figure in square centimetres. 12. Plot the points (1, 3), ( - 3, - 9), and shew that they lie on a line passing through the origin. Name the coordinates of other points on this line. 13. Plot the eight points (0, 5), (3, 4), (5, 0), (4, -3), ( - 5, 0), (0, -5), (-4, 3), (-4, -3), and shew that they are all equidistant from the origin. 14. Plot the two following series of points : (i) (5,0), (5,2), (5,5), (5, -1), (5, -4); (ii) (-4, 8), (-1,8), (0,8), (3,8), (6,8). Shew that they lie on two lines respectively parallel to the axis of y^ and the axis of x. Find the coordinates of the point in which they intersect. GRAPH OF A FUNCTION. O 15. Plot the points (13, 0), {0, - 13), (12, 5), ( - 12, 5), ( - 13, 0), ( - 5, - 12), (5, - 12). Find their locus, (i) by measurement, (ii) by calculation. 16. Plot the points (2, 2), (-3, -3), (4, 4),(-5, -5), shewing that they all lie on a certain line through the origin. Conversely, shew that for every point on this line the abscissa and ordinate are equal. Graph of a Function. 7. Let f{x) represent a function of ^, and let its value be denoted by y. If we give to x a series of numerical values we get a corresponding series of values for y. If these are set off as abscissae and ordinates respectively, we plot a succession of points. If all such points were plotted we should arrive at a line, straight or curved, which is known as the graph of the function fix) ^ or the graph of the equation y-=f{x). The varia- tion of the function for different values of the variable x is exhibited by the variation of the ordinates as we pass from point to point. In practice a few points carefully plotted will usually enable us to draw the graph with sufficient accuracy. 8. The student who has worked intelligently through the preceding examples will have acquired for himself some useful preliminary notions which will be of service in the examples on simple graphs which we are about to give. In particular, before proceeding further he should satisfy himself with regard to the following statements : (i) The coordinates of the origin are (0, 0). (ii) The abscissa of every point on the axis of y is 0. (iii) The ordinate of every point on the axis of x is 0. (iv) The graph of all points which have the same abscissa is a line parallel to the axis of y. {e.g. x = %) (v) The graph of all points which have the same ordinate is a line parallel to the axis of x. {e.g. y = 5.) (vi) The distance of any point F{x, y) from the origin is given by OF^^ = x^+y^. b GRAPHICAL ALGEBRA. Example L Plot the graph oi y = x. When x = 0, y = ; thus the origin is one point on the graph. Also, when x = l, 2, 3, ... -1, -2, -3, ..., y=l, 2, 3, .. -1, -2, -3, ... Thus the graph passes through 0, and represents a series of points each of which has its ordinate equal to its abscissa, and is clearly represented by POP' in Fig. 3. Example 2. Plot the graph oi y = x-]-o. Arrange the values of x and y as follows : x y 3 2 5 1 3 -1 2 -2 -3 6 4 1 By Joining these points we obtain a line MN parallel to that in Example 1. The results printed in larger and deeper type should be specially noted and compared with the graph. They shew that the distances ON, OM (usually called the intercepts on the axefi) are obtained by separ- ately putting x = 0, y = b in the equation of the graph. Note. By observing that in Example 2 each ordinate is 3 units greater than the corresponding ordinate in Example 1, the graph oi y — x + Z may be obtained from that oi y = xhy simply producing each ordinate 3 units in the positive direction. In like manner the equations y = x-\-5, y = x-D represent two parallel lines on opposite sides oi y=x and equi- distant from it, as the student may easily verify for himself. LINEAR GHAPHS. 7 Example 3. Plot the graphs represented by the following equa- tions : (i) y = 2x', (ii) i/ = 2a; + 4; (iii) y = 2x-6. (ii) IT / am. Fig. 4. Here we only give the diagram which the student should verify in detail for himself, following the method explained in the two preceding examples. EXAMPLES II. [/?^ the following examples Nos. 1-18 are arranged in groups of three ; each group should he represented on the same diagram so as to exhibit clearly the position of the three graphs rela- tively to each other. ^ Plot the graphs represented by the following equations : 1. 4. 7. 10. 13. 16. 19. Shew by careful drawing that the three last graphs have a common point whose coordinates are 2, 1. 20. Shew by careful drawing that the equations x-\-y—lO, y=x-4: represent two straight lines at right angles. y = 5x. 2. y = 5x-4. 3. y = 5x + 6. y= -Sx. 5. y= -Sx + S. 6. y=z -3a; -2. y + x = 0. 8. y + x = S. 9. y + 4: = x. 4x = 3y. 11. Sy = 4x + 6. 12. ^y + 3x = S. a;-5 = 0. 14. y-6 = 0. 15. 5y = 6x. Sx + 4cy = 10. 17. 4x + y = 9. 18. 5x-2y = S. 8 GRAPHICAL ALGEBRA. 21. Draw on the same axes the graphs of x = 5, x = 9, ?/ = 3, y = ll. Find the number of units of area enclosed by these lines. 22. Taking one-tenth of an inch as the unit of length, find the area included between the graphs oi x='7, x= -3, y= -2, y = S. 23. Find the area included by the graphs of 2/ = ic + 6, 'i/ = x-6, i/=~-x + 6, y=-x-6. 24. With one millimetre as linear unit, find in square centimetres the area of the figure enclosed by the graphs of i/ = 2a; + 8, y = 2x-S, y= -2x + S, y=-2x-^. 9. The student should now be prepared for the following statements : (i) For all numerical values of a the equation y = ax re- presents a straight line through the origin. (ii) For all numerical values of a and h the equation y = ax-\-h represents a line parallel to y = ax, and cutting of!* an intercept h from the axis of y. 10. Conversely, since every equation involving x and ?/ only in the first degree can be reduced to one of the forms y = ax, y = ax-h b, it follows that every simple equation connecting two variables represents a straight line. For this reason an expression of the form ax + 6 is said to be a linear function of x, and an equation such as y = dx + 6, or ax + by-\-c = 0,i^ said to be a linear equation. Example. Shew that the points (3, -4), (9, 4), (12, 8) lie on a straight line, and find its equation. Assume y = ax + h as the equation of the line. If it passes through the first two points given, their coordinates must satisfy the above equation. Hence -4-:3a + 6, 4 = 9a + 6. 4 These equations give a = ~, b= -S. 4 Hence y = ^x-S, or 4x-^y = 24:, is the equation of the line passing through the first two points. Since x=\2,y = S satisfies this equation, the line also passes through (12, 8). This example may be verified graphically by plotting the line which joins any two of the points and shewing that it passes through the third. APPLICATION TO SIMULTANEOUS EQUATIONS. Application to Simultaneous Equations. 11. It is shewn [E. A., Art. 100] that in the case of a simple equation between x and y, it is possible to find as many pairs of values of ^ and i/ as we please which satisfy the given equation. We now see that this is equivalent to saying that we may find as many points as we please on any given straight line. If, however, we have two simultaneous equations between ^ and ?/, there can only be one pair of values which will satisfy both equations. This is equivalent to saying that two straight lines can have only one common point. Example. Solve graphically the equations : FifS If carefully plotted it will be foiind that these two equations represent the lines in the annexed diagram. On measuring the coordinates of the point at which they intersect it will be found that x = 2, y = S, thus verifying the solution given in E. A. Art. 103, Ex. 1. 12. It will now be seen that the process of solving two simultaneous equations is equivalent to finding the coordinates of the point (or points) at which their graphs meet. 13. Since a straight line can always be drawn by joining am/ two points on it, in solving linear simultaneous equations graphically, it is only necessary to plot two points on each line. The points where the lines meet the axes will usually be the most convenient to select. 10 C4RAPHICAL ALGEBRA. 14. Two simultaneous equations lead to no finite solution if they are inconsistent with each other. For example, the equations ^ + 3y = 2, 3^+9y = 8 are inconsistent, for the second equation can be written .^4-3y = 2|, which is clearly inconsistent with ^ + 3// = 2. The graphs of these two equations will be found to be two parallel straight lines which have no finite point of intersection. Again, two simultaneous equations must be independent. The equations 4r + 3y = l, 16^+12^ = 4 are not independent, for the second can be deduced from the first by multiplying throughout by 4. Thus any 'pair of vahtes which will satisfy one equation will satisfy the other. Graphi- cally these two equations represent two coincident straight lines which of course have an unlimited number of common points. EXAMPLES III. Solve the following equations, in each case verifying the solution graphically : 1. y = 2a; + 3, 2. y = 3aj + 4, 3. y = ^x, y + x = Q. 2/ = a? + 8. 2x-\-y = \%. 4. 2x-y = ^, 5. 3x + 22/=16, 6. Qy-^x = \S, 4a: + 3y = 6. 5^-31/ = 14. ^x = Zy. 7. 2x + y = 0, 8. 2x--y = Z, 9. 2.y = 5ic+15, 10. Prove by graphical representation that the three points (3, 0), (2, 7), (4, -7) lie on a straight line. Where does this line cut the axis oiyt 11. Prove that the three points (1, 1), (-3, 4), (5, -2) lie on a straight line. Find its equation. Draw the graph of this equation, shewing that it passes through the given points. 12. Shew that the three points (3, 2), (8, 8), (-2, -4) lie on a straight line. Prove algebraically and graphically that it cuts the axis of ;r at a distance 1^ from the origin. GRAPHS OF QUADRATIC FUNCTIONS. 11 15. We shall now give some graphs of functions of higher degree than the first. Example I. Plot the graph of 2y = x^. Corresponding values of x and y may be tabulated as follows : X 3 2-5 2 1-5 1 -I _2 -3 y ... 4-5 3125 2 1-125 •5 •5 2 4-5 Here, in order to obtain a figure on a sufficiently large scale, it will be found convenient to take two divisions on the paper for our unit. If the above points are plotted and connected by a line drawn freehand, we shall obtain the curve shewn in Fig. 6. This curve is called a parabola. There are two facts to be specially noted in this example. (i) Since from the equation we have x= ±\/2y, it follows that for every value of the ordinate we have two values of the abscissa, eqnal in magnitude and opposite in sign. Hence the graph is sym- metrical with respect to the axis oi y ; so that after plotting with care enough points to determine the form of the graph in the first quadrant, its form in the second quadrant can be inferred without actually plotting any points in this quadrant. At the same time, in this and similar cases beginners are recommended to plot a few points in each quadrant through which the graph passes. 12 GRAPHICAL ALGEBRA. (ii) We observe that all the plotted points lie above the axis of x. This is evident from the equation ; for since x^ must be positive for all values of x, every ordinate obtained from the equation y = -n must be positive. In like manner the student may shew that the graph of 2y= - x^ is a curve similar in every respect to that in Fig. 6, but lying entirelj^ below the axis of x. Note. Some further remarks on the graph of this and the next example will be found in Art. 21. ^2 Example 2. Find the graph of 2/ = 2ic + —. ■'*^ Here the following arrangement will be found convenient : X 3 2 1 -1 _2 -3 -4 -5 -6 -7 -14 -8 2x 6 4 2 -2 -4 -6 -8 -10 -12 -16 x^ 4 y 2-25 8-25 1 5 •25 •25 1 2-25 4 6 '25 9 12 25 16 2-25 -1-75 -3 -3-75 -4 -3-75 -3 - 1-75 Yl FlgTt From the form of the equation it is evident that every positive value of X will yield a positiv^e value of y, and that as x increases y also increases. Hence the portion of the curve in the first quadrant lies as in Fig. 7, and can be extended indefinitely in this quadrant. In the present case only two or three positive values of x and y need be plotted, but more attention must be paid to the results arising out of negative values of x. MAXIMA AND MINIMA. 13 When i/ = 0, we have — +2^ = ; thus the two values of x in the graph which correspond to y = furnish the roots of the equation 4 /r 16. If f(x) represent a function of ^, an approximate solution of the equation /(x) = may be obtained by plotting the graph of ,y=/(^), and then measuring the intercepts made on the axis of ^. These intercepts are values of ^ which make y equal to zero, and are therefore roots of /(^) = 0. 17. If /(^) gradually increases till it reaches a value a, which is algebraically greater than neighbouring values on either side, a is said to be a maximum value of f{x}. If /(^) gradually decreases till it reaches a value 6, which is algebraically less than neighbouring values on either side, b is said to be a minimum value of f(.v). When y=f{cc) is treated graphically, it is now evident that maximum and minimum values of f(pc) occur at points where the ordinates are algebraically greatest and least in the im- mediate vicinity of such points. Example. Solve the equation x'^-1x-\-\\ = graphically, and find the minimum value of the function x^ -lx+\\. Put y — x^- ^x^- 11, and find the graph of this equation. X 1 2 3 3-6 4 5 6 7 y 11 5 1 -1 -1-25 -1 1 5 11 The values of x which make the function x'^-lx+W vanish are those which correspond to y — 0. By careful measurement it will be found that the intercepts OM and ON are approximately equal to 2-38 and 4-62. The algebraical solution of a:2-7a;+ll=0 gives x = ^{1±sj^). If we take 2*236 as the approximate value of ^5, the values of x will be found to agree with those ob- tained from the graph. -d 14 GRAPHICAL ALGEBRA. Again, x'^-'Jx+M = lx -~] -j. Now (x-^\ must be positive for all real values of x except x = ^, in which case it vanishes, and the value of the function reduces to - -, which is the least value it 4 can have. The graph shews that when x = S-5, y= -1-25, and that this is the algebraically least ordinate in the plotted curve. 18. The following example shews that points selected for graphical representation must sometimes be restricted within certain limits. Example. Find the graph of 0^^ + 1/2 — 36. The equation may be written in either of the following forms : (i) y=^v^36_:r2; (ii) x=i\/36^1/^ F / *i t 5; Fig.9. In order that y may be a real quantity we see from (i) that 36 - x*^ must be positive. Thus x can only have values between - 6 and + 6. Similarly from (ii) it is evident that y must also lie between - 6 and + 6. Between these limits it will be found that all plotted points will lie at a distance 6 from the origin. Hence the graph is a circle whose centre is and whose radius is 6. This is otherwise evident, for the distance of any point F{x,y) from the origin is given by OP = \^x^ + y^. [Art. 8.] Hence the equation x'^-\-y^ = S6 asserts that the graph consists of a series of points all of which are at a distance 6 from the origin. EXAMPLES IV. 15 Note. To plot the curve from equation (ii), we should select a succession of values for y and then find corresponding values of x. In other words we make y the independent and x the dependent variable. The student should be prepared to do this in some of the examples which follow. EXAMPLES IV. 1. Draw the graphs of y = a:^, and x = y'^, and shew that they have only one common chord. Find its equation. 2. From the graphs, and also by calculation, shew that y~-^ cuts a? = -2/2 in only two points, and find their coordinates. 3. Draw the graphs of (i) y'^^-^x', (ii) y = 2x-^', (iii) y^^ + x-2. 4. Draw the graph of y = x + x^. Shew also that it may be deduced from that of y — x^, obtained in Example 1. 5. Shew (i) graphically, (ii) algebraically, that the line y — 2x - 3 x^ meets the curve y = ~j-\-x-2ii\ one point only. Find its coordinates. 6. Find graphically the roots of the following equations to 2 places of decimals : (i) ^+07-2 = 0; (ii) x''--2x = ^', (iii) Ax'^-l&x + ^^O; and verify the solutions algebraically. 7. Find the minimum value of x'^-2x- 4, and the maximum value of 5 + 4a? - 2x'^. 8. Draw the graph of y = [x-\){x-2) and find the minimum value of {x-\){x- 2). Measure, as accurately as you can, the values of X for which {x-\){x~2) is equal to 5 and 9 respectively. Verify algebraically. 9. Solve the simultaneous equations 0:2 + 2/2=100, x + y^U; and verify the solution by plotting the graphs of the equations and measuring the coordinates of their common points. 10. Plot the graphs of x^ + y^ — 2^, 307 + 41/ = 25, and examine their relation to each other where they intersect. Verify the result algebraically. 16 GRAPHICAL ALGEBRA. 19. Infinite and zero values. Consider the X in which the numerator a has a certain fixed value, and the denominator is a quantity subject to change ; then it is clear that the smaller x becomes the larger does the value of the fraction - become. For instance X l^ = lOa, -Y-=:1000a, Y = 1000000a. 10 1000 1000000 By making the denominator x sufficiently small the value of the fraction - can be made as large as we please ; that is, if x is X made less than any quantity that can he named, the value of - X will become greater than any quantity that can he named. A quantity less than any assignable quantity is called zero and is denoted by the symbol 0. A quantity greater than any assignable quantity is called infinity and is denoted by the symbol oo . We may now say briefly when X = 0, the value of - is oo. Again if ^ is a quantity which gradually increases and finally becomes greater than any assignable quantity the fraction becomes smaller than any assignable quantity. Or more briefly when X = cx) , the value of — is 0. ' X 20. It should be observed that when the symbols for zero and infinity are used in the sense above explained, they are subject to the rules of signs which aff'ect other algebraical symbols. Thus we shall find it convenient to use a concise statement such as "when ^= +0, ?/= + oo " to indicate that when a very small and positive value is given to x, the corresponding value of y is very large and positive. 21. If we now return to the examples worked out in Art. 15, in Example 1, we see that when .^= i oc, ?y= +go ; hence the curve extends upwards to infinity in both the first and second quadrants. In Example 2, when x=+»^^ "^-^ ■"-=. '^^ ""Ni^ ^\ ■^i.^ s s. \ 'X s V + ^ T > _i_ _ __ __- o CO o CM X ^^ "^-^^ "^----^ ^ "^^^ "^^^ "^^^ "^^ ^ :::v s S \ \ V \ \ \ y F ± : _ ^ bo E 00 CO 24 GRAPHICAL ALGEBRA. EXAMPLES V. 1. Plot the graph of y = oi^. Shew that it consists of a con- tinuous curve lying in the first and third, quadrants, crossing the axis of X at the origin. Deduce the graphs of (i) y=-o(^; (ii) y = ^7^. 2. Plot the graph of y = x-x^. Verify it from the graphs of y = x, and y = x^. 3. Plot the graph of 2/ = "2' shewing that it consists of two branches lying entirely in the first and second quadrants. Examine and compare the nature and position of the graph as it approaches the axes. 4. Discuss the general character of the graph oi y = —^ where a has some constant integral value. Distinguish between two cases in which a has numerical values, equal in magnitude but opposite in sign. 5. Plot the graphs of (1) y=i+-, (11) 2/-2 + ^. Verify by deducing them from the graphs oi y = -, and y = —^' 6. Plot the graph oi y = x^- So?. Examine the character of the curve at the points (1, -2), ( - 1, 2), and shew graphically that the roots of the equation x'^-Sx = are approximately -1732, 0, and 1-732. 7. Solve the equations : Sx-\-2y=l6, xy=lO, and verify the solution by finding the coordinates of the points where their graphs intersect. 8. Plot the graphs of ... 15 -x^ .... 10-2/2 (1) y=~^. (n) ^ = -^' and thus verify the algebraical solution of the equations x'^ + xy = l5i y^ + xy = 10. EXAMPLES V. 25 9. Trace the curve whose equation is y = ^ . shewing that it has two branches, one lying in the first and third quadrants, and the other entirely in the fourth. Find the equations of its asymptotes. Plot the graphs of 10. l+x 2^ = 1-^- 11. l+x^ y-i-x' 12. 13. {x-\){x-2) y x-z 14. y~x^-x+l' 15. x'^ + bx^-Q y ~ x^^\ 16. y=x^-6x'^+Ux-6. 17. \^y = x^-^x^^x-^. 18. 20 1Q 40:r 2^-^2 + 10 ■ »■ y'li^ 21. {x-2){x-3) y x-5 ' 22. {x-\){x-%{x^\) y 4 23. 2/2 = a:2-5a^ + 4. 24. 42/2 = 0^2 (5-0?). _ x{^-x){x-^) _ {x + 7){x-4){x-l0) ^- y~ ^^T5 ^^- y~ x'+5 _.t2( 49_^2) ( 81 -x2) (0:2 -4) ^'- 2/ -— ^0 ^^' y - loo • 29. 52/3 = o;(o;2-64). 30. 5i/3=.aj2(36-o:2). 31. Plot the graphs of y = x^, and of y = 2x'^ + x -2. Hence find the roots of the equation x^ - 2x^ - o; + 2 = 0. 32. Find graphically the roots of the equation o:3-4o:2-5o; + 14 = to three significant figures. 26 GRAPHICAL ALGEBRA. 29. Besides the instances already given there are several of the ordinary processes of Arithmetic and Algebra which lend themselves readily to graphical illustration. For example, the graph of y = x^ may be used to furnish numerical square roots. For since x=sly^ each ordinate and corresponding abscissa give a number and its square root. Similarly cube roots may be found from the graph oiy=o(^. Exam-pie 1. Find graphically the cube root of 10 to 3 places of decimals. The required root is clearly a little greater than 2. Hence it will be enough to plot the graph of y — x^ taking aj = 2'l, 2*2, ... The corresponding ordinates are 9-26, 10 '65, ... When x = % y = ^. Take the axes through this point and let the units for x and y be 10 inches and '5 inch respectively. On this scale the portion of the graph differs but little from a straight line, and yields results to a high degree of accuracy. 11 Y y ' , / / y 10 / A y y I / ' 9 y ^ y ^ 1 y y y I 8 ^ y 2-1 2-154 2-2 X When 2/ = 10, the measured value of x will be found to be 2*1 54. Example, 2. Shew graphically that the expression 4^2 + 4:r - 3 is negative for all real values of x between '5 and - 1 '5, and positive for all real values of x outside these limits. [Fig. 16.] Put 2/ = 4:r2 + 4x - 3, and proceed as in the example given in Art. 16, taking the unit for x four times as great as that for y. It will be found that the graph cuts the axis of x at points whose abscissae are '5 and - 1 '5 ; and that it lies below the axis of x between these points. That is, the value of y is negative so long as x lies between •5 and - 1 '5, and positive for all other values of x. ILLUSTRATIVE EXAMPLES. 27 Or we may proceed as follows : Put y^ = 4x^, and 2/2= -4aT + 3, and plot the graphs of these two equations. At their points of intersection 2/1 = 2/0? and the values of X at these points are found to be '5 and - 1 -5. Hence for these values of x we have 4^2 ^ _ 4^ + 3^ or 4a:2 + 4a; - 3 = 0. Thus the roots of the equation 407^ + 4a7 - 3 — are furnished by the abscissse of the common points of the graphs of ix"^ and -4x + 3. Again, between the values '5 and - I'S for x it will be found graphically that 2/1 is less than yo, hence i/j - 2/2' ^^ ^^^ + 4:r - 3 is negative. 1 P Y \ ^ / \ \ / ^ \ / \ 5 1 \ \ 1 1 \ \, \ \ \ \ \ 1/ \ l/f X' V ^ ^ X - 1-5 v -1 / \ 1 1-5 \ / ^ \ / V y Y Fig. 16. Both solutions are here exhibited. The upper curve is the graph of y — \x^ ; PQ is the graph of y— - 4ir + 3 ; and the lower curve is the graph oi y — 4x'^ + 4a: - 3. 30. Of the two methods in the last Example the first is the more direct and instructive ; but the second has this advantage : If a number of equations of the form a;'^=pj!J-hg have to be solved graphically, y—x^ can be plotted once for all on a con- venient scale, and y=px-^q can then be readily drawn for different values of p and q. Equations of higher degree may be treated similarly. 28 GRAPHICAL ALGEBRA. For example, the solution of such equations as x^=px + q^ or a^ = ax^ + hx-\-c can be made to depend on the intersection of y=^x^ with other graphs. Example. Find the real roots of the equations (i) ic»-2-5ic-3 = 0; (ii) a^-^x + 2=0. Here we have to find the points of intersection of (i) y^x^, 2/ = 2-5a; + 3; (ii) y = x\ 2/ = 3aj-2. Plot the graphs of these equations, choosing the unit for x five times as great as that for y. X' r ^ -^yy-k-s r^+J Y' Fi^. 17. It will be seen that 2/ = 2'5a: + 3 meets y = x^ only at the point for which x = 2. Thus 2 is the only real root of equation (i). Again y = ^x-2 touches y — x^ at the point for which x = \, and cuts it where x= -2. Corresponding to the former point the equation cr^ - 3x + 2 = has two equal roots, Thus the roots of (ii) are 1, 1, -2. TRIGONOMETRICAL FUNCTIONS. 29 31. Apart from questions of convenience with regard to any particular graph, we may observe that in many cases the variables whose values are plotted on the two axes denote magnitudes of different kinds, so that there is no necessary relation between the units in which they are measured. A good illustration of this kind is furnished by tracing the variations of the Trigonometrical functions graphically. Example. Trace the graph of sin x. In any work on Trigonometry it is shewn that as the angle x increases from 0° to 90°, the value of sin x is positive, and increasing gradually from to 1. From 90° to 180°, sin a? is positive, and decreasing from 1 to 0. From 180° to 270°, sin x is negative, and increasing numerically from to - 1 . And from 270° to 360°, sin x is negative, and decreasing numerically from - 1 to 0. (See Hall and Knight's Elementary Trigonometry, Art. 86. ) We shall here exhibit these variations independently by putting 2/ = sin X, and plotting the values of y corresponding to values of ic differing by 30°. By the aid of a table of sines we have : X 0° 30° 60° 90° 120° 150° 180° 210° 240° 270° ... or -5 -866 1 '866 sin a? •5 -5 -866 -1 ... The graph is represented by the Fig. 18. continuous waving line shewn in 1::: '--":":-;: : :: : . - ' i. .... (L _ _ -^--- :z:"_ - ._..s. . _: ::z .: __: _ : : : : .. s^. : : : : - _ "::~:: :_:_: z -I . . - - _ .s. - - - - _ d ^:::-&lo'^::p^:-bo^:-:20| 150^-^8 qo U.y \^t(fr ._j^__ __ 7" ' :::==:;: i": :::"::^^_' " ~ .^ :r-»,-_-=::: Fig. i8. On the rr-axis each division represents 6°, and on the ^-axis ten divisions have been taken as the unit. C 30 GRAPHICAL ALGEBRA. EXAMPLES VI. 1. Draw the graph oi y = x^ on o, scale twice as large as that in Fig. 13, and employ it to find the squares of 72, 1'7, 3 '4; and the square roots of 7*56, 5*29, 9'6L 2. Draw the graph of y=^sjx taking the unit for y five times as great as that for x. By means of this curve check the values of the square roots found in Example 1. 3. From the graph of y = x^ (on the scale of the diagram of Art. 29) find the values of ^^^9 and ^'^'^ to 4 significant figures. 4. A boy who was ignorant of the rule for cube root required the value of Vl4'71. He plotted the graph of y = x^, using for x the values 2*2, 2*3, 2*4, 2*5, and found 2*45 as the value of the cube root. Verify this process in detail. From the same graph find the value of \/13-8. 5. Find graphically the values of x for which the expression ic^ - 2a; - 8 vanishes. Shew that for values of x between these limits the expression is negative and for all otlier values positive. Find the least value of the expression. 6. From the graph in the preceding example shew that for any value of a greater than 1 the equation x'^ -2x^-a = cannot have real roots. 7. Shew graphically that the expression x'^-Ax + 1 is positive for all real values oi x. 8. On the same axes draw the graphs of 2/ = a;^, y = x + Q, y = x-6, y=-x + Qj y=-x-6. Hence discuss the roots of the four equations x'^-x-Q = 0, x^-x + Q = 0, x'^ + x-6 = 0, x'^ + x + Q = 0. 9. If X is real, prove graphically that 5 - 4a: - cc^ is not greater than 9 ; and that 4:X^ - 4a: + 3 is not less than 2. Between what values of x is the first expression positive ? 10. Solve the equation x!^ = Sx'^ + 6x -H graphically, and shew that the function x^ - Sx^ - 6a: + 8 is positive for all values of x between -2 and 1, and negative for all values of x between 1 and 4. 11. Shew graphically that the equation x^-\-px + q = has only one real root when p is positive. EXAMPLES VI. 31 12. Trace the curve whose equation is y = 2^. Find the approximate values of 2^'^^ and 2^^'^. Express 12 as a power of 2 approximately. Prove also that log2 26 "9 + log^ 38 = 10. 13. By repeated evolution find the values of 10^, 10^, 10^, 10^^^. By multiplication find the values of 10^'^, 10^'^, 10^^^,10^X10^^. Use these values to plot a portion of the curve y = \0^ on a large scale. Find correct to three places of decimals the values of log 3, log 1*68, log 2-24, log 34*3. Also by choosing numerical values for a and 6, verify the laws log ah = log a + log h ; log t = log a - log h. [By using paper ruled to tenths of an inch, if 10 in. and 1 in. he taken as units for x and y respectively, a diagonal scale will give values ofx correct to three decimal places and values of y correct to tivo.'] 14. Calculate the values of a:(9 - xY for the values 0, 1, 2, 3, ... 9 of X. Draw the graph of a; (9 - a; )^ from a: = to a: = 9. If a very thin elastic rod, 9 inches in length, fixed at one end, swings like a pendulum, the expression x{^-xf measures the tendency of the rod to break at a place x inches from the point of suspension. From the graph find where the rod is most likely to break, 15. If a man spends 22s. a year on tea whatever the price of tea is, what amounts will he receive when the price is 12, 16, 18, 20, 24, 28, 33, and 36 pence respectively ? Give your results to the nearest quarter of a pound. Draw a curve to the scale of 4 lbs. to the inch and 10 pence to the inch, to shew the number of pounds that he would receive at intermediate prices. 16. Draw the graphs of cos x and tan x, on a scale twice as large as that in Art. 31. 17. Draw the graph of sin x from the following values of x : 5°, 15°, 30°, 45°, 60°, 75°, 85°, 90°. Find the value of sin 37°, and the angle whose sine is '8. 18. Find from the tables the value of cos x when x = {)% 10°, 20°, 30°, 40°, 50°, 60°. Draw a curve on a large scale shewing how cos x varies as x increases from 0° to 60°. Find from the curve the values of cos 25° and cos 45°. Verify by means of the tables. 19. Draw on the same diagram the graphs of the functions sin x, cos x, and sin x + cos x. Derive from the figure the general solution of sin a? + cos a? = 0. 32 GRAPHICAL ALGEBRA. 20. The range of a certain gun is 1000 sin 2^ yards, where A is the elevation of the gun. Find from the tables the value of 1000 sin 2A when A has the values 10°, 15°, 20°, 25°, 30°, 35°, 40°, 45°, 50°, and draw a curve shewing how the range v^aries as A increases from 10° to 50°. 21. From the tables find the values of tan 10:r-2 tan 9a:; + l for the following values of x : 0°, 1°, 2°, ... 9°. Draw a curve shewing how tan lOic - 2 tan 9a; + 1 varies with x when x lies between 0° and 9°. Find to the nearest tenth of a degree a value of x for which the given expression vanishes. Practical Applications. 32. In all the cases hitherto considered the equation of the curve has been given, and its graph has been drawn by first selecting values of a: and ?/ which satisfy the equation, and then drawing a line so as to pass through the plotted points. We thus determine accurately the position of as many points as we please, and the process employed assures us that they all lie on the graph we are seeking. We could obtain the same result without knowing the equation of the curve provided that we were furnished with a sufficient number of corresponding values of the variables accurately calculated. Sometimes from the nature of the case the form of the equa- tion which connects two variables is known. For example, if a quantity y is directly proportional to another quantity x it is evident that we may put y — ax^ where a is some constant quantity. Hence in all cases of direct proportionality between two quantities the graph which exhibits their variations is a straight line through the origin. Also since two points are sufficient to determine a straight line, it follows that in the cases under consideration we only require to know the position of one point besides the origin, and this will be furnished by any pair of simultaneous values of the variables. Example 1. Given that 5*5 kilograms are roughly equal to 12*125 pounds, shew graphically how to express any number of pounds in kilograms. Express 7| lbs. in kilograms, and 4^ kilograms in pounds. PRACTICAL APPLICATIONS. 33 Here measuring pounds horizontally and kilograms vertically, the required graph is obtained at once by joining the origin to the point whose coordinates are 12*125 and 5 '5. Y| - L -tn y _ -— -^ -^ 4- 2E ^ ) ^ ~J^~ B-^ k^ s5 -- T ^ 1: — " h )uha s _ _ _ _ _ __ 5 10 12-125 X Fi^. 19. By measurement it will be found that 7^ lbs. =3*4 kilograms, and 4^ kilograms = 9 -37 lbs. Example 2. The expenses of a school are partly constant and partly proportional to the number of boys. The expenses were £650 for 105 boys, and £742 for 128. Draw a graph to represent the expenses for any number of boys ; find the expenses for 115 boys, and the number of boys that can be maintained at a cost of £710. If the expenses for x boys are represented by £y, it is evident that X and y satisfy a linear equation y = ax + h, where a and h are constants. Hence the graph is a straight line. 800 700 600 Y U ^ — 1 -^ ^ 1 — "" n -^ -' " ^ p- -1 "^ " ■ _j _ L- __j 100 105 110 115 Fig". 20. 120 128 X As the numbers are large, it will be convenient if we begin measuring ordinates at 600, and abscissne at 100. This enables us to bring the requisite portion of the graph into a smaller compass. The points P and Q are determined by the data of the question, and the line PQ is the graph required. By measurement we find that when .t=115, i/ = 690; and that when i/ = 710, a;=120. Thus the required answers are £690, and 120 boys. 34 GRAPHICAL ALGEBRA. 33. Sometimes corresponding values of two variables are obtained by observation or experiment. In such cases the data cannot be regarded as free from error ; the position of the plotted points cannot be absolutely relied on ; and we cannot correct irregularities in the graph by plotting other points selected at discretion. All we can do is to draw a curve to lie as evenly as possible among the plotted points, passing through some perhaps, and with the rest fairly distributed on either side of the curve. As an aid to drawing an even continuous curve a thin piece of wood or other flexible material may be bent into the requisite curve, and held in position while the line is drawn.* When the plotted points lie approximately on a straight line, the simplest plan is to use a piece of tracing paper or celluloid on which a straight line has been drawn. When this has been placed in the right position the extremities can be marked on the squared paper, and by joining these points the approximate graph is obtained. Example 1. The following table gives statistics of the population of a certain country, where P is the number of millions at the beginning of each of the years specified. Year 1830 1835 1840 1850 1860 1865 1870 1880 P 20 22-1 23-5 29-0 34-2 38-2 41-0 49-4 Let t be the time in years from 1830. Plot the values of P vertically and those of t horizontally and exhibit the relation between P and ^ by a simple curve passing fairly evenly among the plotted points. Find what the population was at the beginning of the years 1848 and 1875. The graph is given in Fig. 21 on the opposite page. The popula- tions in 1848 and 1875, at the points A and B respectively, will be foimd to be 27 "8 millions and 45 '3 millions. Example 2. Corresponding values of x and y are given in the following table ; X 1 4 6-8 8 9-5 12 14-4 y 4 8 12-2 13 15-3 20 24-8 Supposing these values to involve errors of observation, draw the graph approximately and determine the most probable equation between x and y. [See Fig. 22 on p. 36.] * One of " Brooks' Flexible Curves " will be found very useful. PRACTICAL Applications. 35 r \ ( ( \ \ \ "^ jS 0Q\ -t X N S \ A \ -^ K \ "N "^^_4_ \ ^ V ^ ^ i_ A \ A V " A ^ ,!v_ 5 ^ 4^ V. 5 v_ 5 V x "N i: ^ _5 V 5 ~^V 3 \_ Jl Y -\ 00 bfl o o o CO s- GRAPHICAL ALGEBRA. After carefully plotting the given points we see that a straight line can be drawn passing through three of them and lying evenly among the others. This is the required graph. 20 10 Y Fig. 22. 10 Assuming y = ax-\-h for its equation, we find the values of a and h by selecting two pairs of simultaneous values of x and y. Thus substituting a; = 4, 2/ = 8, and x=\2,y = 20 in the equation, we obtain a = l*5, 6 = 2. Thus the equation of the graph is y-—\'^x + 2. 34. In the last Example as the graph is linear it can be produced to any extent within the limits of the paper, and so any value of one of the variables being determined, the corre- sponding value of the other can be read off. When large values are in question this method is not only inconvenient but unsafe, owing to the fact that any divergence fron) accuracy in the portion of the graph drawn is increased when the curve is produced beyond the limits of the plotted points. The follow- ing Example illustrates the method of procedure in such cases. Example. In a certain machine P is the force in pounds required to raise a weight of W pounds. The following corresponding values of P and W were obtained experimentally : p 3 08 3-9 6-8 8-8 9-2 * 11 13-3 w 21 36-25 66-2 87-5 103-75 120 152-5 By plotting these values on squared paper draw the graph con- necting P and W, and read off the value of P when W=10. Also determine a linear law connecting P and W ; find the force necessary to raise a weight of 310 lbs., and also the weight which could be raised by a force of 180*6 lbs. PRACTICAL APPLICATIONS. 37 As the page is too small to exhibit the graphical work on a convenient scale we shall merely indicate the steps of the solution, which is similar in detail to that of the last example. Plot the values of P vertically and the values of W horizontally. It will be found that a straight line can be drawn through the points corresponding to the results marked with an asterisk, and lying evenly among the other points. From this graph we find that when W=10, P = l. Assume P = aPF+?>, and substitute for P and W iroxn. the values corresponding to the two points through which the line passes. By solving the resulting equations we obtain a = '08, 6 = I '4. Thus the linear equation connecting P and W is P= '08 W^+ 1*4. This is called the Law of the Machine. From this equation, when W=^\0, P = 2Q% and when P= 180*6, ff = 2240. Thus a force of 26 '2 lbs. will raise a weight of 310 lbs.; and when a force of 180*6 lbs. is applied the weight raised is 2240 lbs. or 1 ton. Note. The equation of the graph is not only useful for determin- ing results difficult to obtain graphically, but it can always be used to check results found by measurement. 35. The example in the last article is a simple illustration of a method of procedure which is common in the laboratory or workshop, the object being to determine the law connecting two variables when a certain number of simultaneous values have been determined by experiment or observation. Though we can always draw a graph to lie fairly among the plotted points corresponding to the observed values, unless the graph is a straight line it may be difi&cult to find its equation except by some indirect method. For example, suppose x and y are quantities which satisfy an equation of the form xy = ax + hy, and that this law has to be discovered. By writing the equation in the form a h ^ , _ y X where u = -^ 'V = ~, it is clear that u, v satisfy the equation of a straight line. In other words, if we were to plot the points corresponding to the reciprocals of the given values, their linear connection would be at once apparent. Hence the values of a and h could be found as in previous examples, and the required law in the form xy = ax + hy could be determined. 38 (^RA1*HICAL ALGEBRA. Again, suppose x and ^ satisfy an equation of the form x^^ = c, where tz. and c are constants. By taking logarithms, we have n log a^ + log y = log a The form of this equation shews that log ^ and log ^ satisfy the equation to a straight line. If, therefore, the values of log.r and logy are plotted, a linear graph can be drawn, and the constants n Siud c can be found as before. Example. The weight, y grammes, necessary to produce a given deflection in the middle of a beam supported at two points, x centi- metres apart, is determined experimentally for a number of values of X with results given in the following table : X 50 60 70 80 90 100 y 270 150 100 60 47 32 log a: logy 1-699 2-431 1-778 2-176 1-845 2 000 1-903 1-778 1-954 1-672 2-000 1 519 Assuming that x and y are connected by the equation x^^y — c^ find n and c. From a book of tables we obtain the annexed values of log x and log y corresponding to the observed values of x and y. By plotting these we obtain the graph given in Fig. 23, and its equation is of the form n log X + log y = log c. To obtain n and c, choose two extreme points through which the line passes. It will be found that when log x = l -642, log y = '2'Q and when loga^ = 2'l, log2/=r21. Substituting these values, we have 2-6 + »i X 1-642:= log c (i), l-21-f-wx2-l =logc (ii); .-. 1-39 -0-45891 = 0; whence ?i = 3-04. .-. from (ii) log c = 6 -38 + 1 '21 = 7-59; .*. c = 39 X 10^ from the tables. Thus the required equation is x^y = S9 x 10^ The student should work through this example in detail on a larger scale. The adjoining figure was drawn on paper ruled to tenths of an inch and then reduced to half the original scale. EXAMPLES VII. 41 7. At different ages the mean after-lifetime ("expectation of life") of males, calculated on the death rates of 1871-1880, was given by the following table : Age 6 10 14 18 22 26 27 Expectation 50-38 47-60 44-26 40-96 37-89 34-96 34-24 Draw a graph to shew the expectation of any male between the ages of 6 and 27, and from it determine the expectation of persons aged 12 and 20. 8. In the Clergy Mutual Assurance Society the premium (£P) to insure £100 at different ages is given approximately by the following table : Age 20 22 25 30 35 40 45 50 55 P 1-8 1-9 2-0 2-3 2-7 31 3-6 4-4 5-5 Illustrate the same statistics graphically, and estimate to the nearest shilling the premiums for persons aged 34 and 43. 9. If W is the weight in ounces required to stretch an elastic string till its length is I inches, plot the following values of W and I : w 2-5 3-75 6-25 7-5 10 11-25 I 8-5 8-7 9-1 9-3 9-7 9-9 From the graph determine the unstretched length of the string, and the weight the string will support when its length is 1 foot. 10. In the following table P and A (expressed in hundreds of pounds) represent the Principal and corresponding Amount for 1 year at 3 per cent, simple interest. p 2-3 2-7 3-0 3-5 3-9 5-2 7-6 A 2-369 2-781 3-090 3-605 4017 5-356 7-828 Plot the values of P and ^ on a large scale, and from the graph determine the Principal which will amount to (i) £329. 12s. ; (ii) £597. 8s. 42 GRAPHICAL ALGEBRA. 11. The highest and lowest marks gained in an examination are 297 and 132 respectively. These have to be reduced in such a way that the maximum for the paper (200) shall be given to the first candidate, and that there shall be a range of 150 marks between the first and last. Find the equation between x, the actual marks gained, and y, the corresponding marks when reduced. Draw the graph of this equation, and read off the marks which should be given to candidates who gained 200, 262, 163 marks in the examination. 12, A body starting with an initial velocity, and subject to an acceleration in the direction of motion, has a velocity of v feet per second after t seconds. If corresponding values of v and t are given by the annexed table. V 9 13 17 21 25 29 33 37 41 45 t 1 2 3 4 5 6 7 8 9 10 plot the graph exhibiting the velocity at any given time. Eind from it (i) the initial velocity, (ii) the time which has elapsed when the velocity is 28 feet per second. Also find the equation between V and t. 13. The connection between the areas of equilateral triangles and their bases (in corresponding units) is given by the following table : Area •43 1-73 3-90 6-93 10-82 15-59 Base 1 2 3 4 5 6 Illustrate these results graphically, and determine the area of an equilateral triangle on a base of 2*4 ft. 14. A body falling freely under gravity drops s feet in t seconds from the time of starting. If corresponding values of s and t at intervals of half a second are as follows : t •5 1 1-5 2 2-5 3 3-5 4 s 4 16 36 64 100 144 196 256 draw the curve connecting s and t, and find from it (i) the distance through which the body has fallen after 1 -8". (ii) the depth of a well if a stone takes 3*16" to reach the bottom. EXAMPLES VII. 43 15, A body is projected with a given velocitj'^ at a given angle to the horizon, and the heiglit in feet reached after t seconds is given by the equation ^ = 64^ - 16^^. Find the values of h at intervals of ^th of a second and draw the path described by the body. Find the maximum value of h, and the time after projection before the body reaches the ground. 16. The keeper of a hotel finds that when he has G guests a day his total daily profit is P pounds. If the following numbers are averages obtained by comparison of many days' accounts determine a simple relation between P and G. G 21 27 29 32 35 P -1-8 2 3-2 4-5 6-6 For what number of guests would he just have no profit ? 17. A man wishes to place in his catalogue a list of a certain class of fishing rods varying from 9 ft. to 16 ft. in length. Four sizes have been made at prices given in the following table : 9 ft. 11 ft. 9 in. 14 ft. 4 in. 16 ft. 15.9. 22.9. 31s. 38.S. Draw a graph to exhibit prices for rods of intermediate lengths, and from it determine the probable prices for rods of 13 ft. and 15 ft. 8 in. 18. The following table gives the sun's position at 7 a.m. on different dates : Mar. 23 Ap. 3 Ap.20 Mays May 27 June 22 July] 8 Aug. 5 Aug. 25 80° E. 82° E. 85° E. 89° E. 92° E. 95° E. 94° E. 91° E. 85° E. Shew these results graphically, and estimate approximately the sun's position at the same hour on June 8th. 19. At a given temperature p lbs. per square inch represents the pressure of a gas which occupies a volume of v cubic inches. Draw a curve connecting p and v from the following table of corresponding values : V 36 30 25-7 22-5 20 18 16-4 15 V 5 6 7 8 9 10 11 12 44 GRAPHICAL ALGEBRA. 20. Plot on squared paper the following measured values of x and y, and determine the most probable equation between x and y : X 3 5 8-3 11 13 15-5 18-6 23 28 y 2 2-2 3-4 3-8 4 4-6 5-4 6-2 7-25 21. The following table refers to aqueous solution of ammonia at a given temperature ; x represents the specific gravity of the solu- tion, and y the percentage of ammonia : x -996 •992 •988 •984 •980 •976 •968 y •91 184 2^80 3^80 4 80 5^80 7^82 Draw a graph shewing the variations of x and y, and find its equation. 22. Corresponding values of x and y are given in the following table : X 1 3-1 6 9^5 12-5 16 19 23 y 2 2^8 4 2 5 3 6-6 8-3 9 10-8 Supposing these values to involve errors of observation, draw the graph approximately, and determine the most probable equation between x and y. Find the correct value of y when a: =19, and the correct value of x when 2/ = 2 •8. 23. The following corresponding values of x and y were obtained experimentally : X 0^5 1-7 3-0 47 57 7^1 8^7 9 9 10-6 ir8 y 148 186 265 326 388 436 529 562 611 652 It is known that they are connected by an equation of the form y = ax-\-b, but the values of x and y involve errors of measurement. Find the most probable values of a and b, and estimate the error in the measured value of y when x = 9'9. EXAMPLES VII. 45 24. In a certain machine P is the force in pounds required to raise a weight of W pounds. The following corresponding values of P and W were obtained experimentally : p 2-8 3-7 4-8 5-5 Q'b 7-3 8 9-5 10-4 11-75 w 20 25 31-7 35-6 45 52-4 57-5 65 71 82-5 Draw the graph connecting P and W, and read off the value of P when ^=60. Also determine the law of the machine, and find from it the weight which could be raised by a force of 31 "7 lbs. 25. The following values of x and y, some of which are slightly inaccurate, are connected by an equation of the form y^ax^ + h. X 1 1-6 3 3-7 4 5 5-7 6 6-3 7 y 3-25 4 5 6-5 7-4 9-25 10-5 11-6 14 15-25 By plotting these values draw the graph, and find the most probable values of a and h. Find the true value of x Wiien 3/ = 4, and the true value of y when x — Q. 26. The following table gives cot responding values of two variables X and y : X 2-75 3 3-2 3-5 4-3 4-5 5-3 6 7 8 10 y 11 9-8 8 6-5 6-1 5-4 5 4-3 4-1 4 3-9 These values involve errors of observation, but the true values are known to satisfy an equation of the form xy = ax-\-by. Draw the graph by plotting the points determined by the above table, and find the most probable values of a and h. Find the correct values of y corresponding to x = 3'5, and x = 7. 27. Observed values of x and y are given as follows : X 100 90 70 60 50 40 y 30 31-08 33-5 35-56 37-8 40-7 Assuming that x and y are connected by an equation of the form xy"^ = c, find n and c. 46 GRAPHICAL ALGEBRA. 28. The following values of x and y involve errors of observation : X 66-83 63-10 58-88 51-52 48-53 44-16 40-36 37-15 y 144-5 158-5 177-8 208-9 236 264-9 309 346-7 If X and y satisfy an equation of the form x'^y = c, find n and c. 29. In the following table the values of C and C represent the calculated and observed amounts of water, in cubic feet per second, flowing through a circular orifice for diflpereut heads of water repre- sented by H feet. H 60 69-12 82 9216 106 115-2 134 C •0133 -0141 -0154 -0163 -0175 •0182 -0197 G' •0133 •0141 •0153 •0162 •0173 -0180 •0194 Plot the graph of G and H and also that of G' and H, and deduce the probable error in the observed flow for a head of 120 feet. 30. The following table gives the pressures (in lbs. per sq. in.) and corresponding Fahrenheit temperatures at which water boils : p 29-7 24-54 17-53 14-^ 12-25 9-80 7-84 t 249-6 239-0 221-0 2120 203-0 192-3 182-0 Shew graphically the relation between temperature and pressure of boiling water. 31. It is known that the relation of pressure to volume in satu- rated steam under certain conditions is of the form jsz;** = constant. Find the value of the index n from the following data : p 10-2 14-7 20-8 24-5 33-7 39-2 45-5 V 37-5 26^6 19-2 16-4 12-2 10-6 9-2 where /> is measured in lbs. per sq. in., and v is the volume of 1 lb. of steam in cub. ft. EXAMPLES VII. 47 32. The following table gives the speed and corresponding indi- cated horse-power of the engines of a ship : Speed in knots 11 12 4 13-3 14-25 14-8 15-5 I.H.R 1000 1500 2000 2500 3000 3500 At what speed will she go when she develops 4000 I.H.P. ? 33. In testing a steam-engine when steam was expanded to 4*8 times its original volume, the following quantities of steam per indicated horse-power per hour were used : Steam per I.H.P. per hr. in lbs. 16-9 17 17-2 18 20-3 I.H.P. 40-5 33 25-5 19 11 When the ratio of expansion in the engine was 10 instead of 4*8, the steam used was as follows : Steam per I.H.P. per hr. in lbs. 15 15-5 16 18 26-5 LH.P. 33 27 2 23 15 5 At what H.P. will the consumption of steam be the same in the two cases, and what is the consumption of steam at that H.P. ? 34. The power required to produce a given speed in the case of each of two ships is given in the following tables : (ii) (i) Speed 8 107 12-7 14 16 16-2 I.H.P. 500 1000 1500 1950 2800 3000 Speed 8 10 12 12-5 13-5 14-5 161 16-7 I.H.P. 1 200 400 920 1100 1500 2000 3000 { 3500 At what speed will they generate the same H.P. ? 48 GRAPHICAL ALGEBRA. 35. £^ represents the amount at compound interest of £P in n years at r per cent. ; the following table gives corresponding values of A and n : A 218-5 238-6 268-5 310-5 360-6 417-8 n 3 6 10 15 20 25 By plotting the values of n and log^, determine a simple alge- braical relation between them. Thence find the numerical values of P and r ; also from the graph find in how many years £200 will amount to £301. Qs. at 3 per cent. 36. The keeper of a restaurant finds that when he has G guests a day his total daily expenditure is E pounds, and his total daily receipts amount to R pounds. The following numbers are averages obtained by comparison of his books on many days : G 210 270 320 360 E 16-7 19-4 21-6 23-4 R 15 8 21-2 26-4 29-8 By plotting these values find E and R when he has 340 guests. What number of guests per day gives him (i) no profit, (ii) £6 profit? Find simple algebraical relations between E and G, R and (r, P and G, where £P is the daily profit. 37. A manufacturer finds that when he is employing W workmen his total weekly expenditure (including wages, material, coal, gas, insurance of premises, etc. ) amounts to E pounds, and his receipts amount to R pounds. After carefully balancing his books for many weeks, the following table of average results was drawn up : w 25 30 35 45 50 60 E 30 2 35-4 40-1 48-0 53-1 59-8 R 27-5 39-1 49-8 75-0 84-8 109-2 From these data determine simple algebraical relations between E and W, R and W, P and ff , where £P represents his weekly profits. Also find graphically (i) the number of workmen necessary to ensure a weekly profit of £18. 10s., (ii) the smallest number of men that will enable the manufacturer to pay expenses. EXAMPLES VII. 49 38. At the following draughts in sea water a particular vessel has the following displacements : Draught h feet 15 12 9 6-3 Displacement T tons 2098 1512 1018 586 By plotting log J' and log A on squared paper, obtain a simple relation between T and h. If one ton of sea water measures 35 cubic feet find the relation between V and h, if V is the displace- ment in cubic feet. 39. The following quantities are thought to follow a law of the form^v" = c. V 1 2 3 4 5 V 205 114 80 63 52 Ascertain if this is the case, and find the most probable values of n and c. 40. In some experiments in towing a canal boat the following observations were made ; P being the pull in pounds and v the speed of the boat in miles per hour : p 76 160 240 320 370 V 1-68 2-48 3-18 3-60 4 03 By plotting logP and logv, shew that P and v approximately satisfy an equation of the form P = hv^, and find the best values for a and h. 41. In the following table h represents the draught in feet of a certain vessel when her displacement is D tons. h 6 10 12 15 D 407 998 1343 2070 By plotting the values of log h and log D determine an algebraical relation between D and h. Find the displacement for a draught of 8 ft. 50 GRAPHICAL ALGEBRA. Miscellaneous Applications of Linear Graphs. 36. When two quantities .r and y are so related that a change in one produces a proportional change in the other, their variations can always be expiessed by an equation of the form y = ax^ where a is some constant quantity. Hence in all such cases the graph which exhibits their variations is a straight line through the origin^ so that in order to draw the graph it is only necessary to know the position of one other point on it. Such examples as deal with work and time, distance and time (when the speed is uniform), quantity and cost of material, principal and simple interest at a given rate per cent., may all be illustrated by linear graphs through the origin. Example 1. At 8 a.m. A starts from P to ride to Q which is 48 miles distant. At the same time B sets out from Q to meet A. If A rides at 8 miles an hour, and rests half an hour at the end of every hour^ while B ivalks uniformly at 4 miles an hour, find graphically (i) the time and place of meeting ; (ii) the distance between A and B at II a.m.; (iii) at what time they are 14 miles apart. In Fig. 24, on the opposite page, let the position of P be chosen as origin ; let time be measured horizontally from 8 a.m. (1 inch to 1 hour), and let distance be measured vertically (1 inch to 20 miles). In 1 hr. A rides 8 mi.; therefore the point D (1, 8) marks his position at 9 a.m. In the next half -hour he makes no advance towards Q ; therefore the corresponding portion of the graph is DE. The details of ^'s motion may now be completed by the broken line PDEFGHKX. On the vertical axis mark PQ to represent 48 mi. and mark the hours on the horizontal line through Q. At 9 a.m. B has walked 4 mi. towards P. Measuring a distance to represent 4 mi. doivn- ivards we get the point R, and QR produced is the graph of B's motion. It cuts ^'s graph at X. Hence the point of meeting is X, which is 28 mi. from P, and the time is 1 p.m. The distance between A and B at any time is shewn by the difference of the ordinates. Thus at 11 a.m. their distance apart is MG, which represents 20 mi. Lastly, NT represents 14 mi.; thus A and B are 14 mi. apart at 11.30 a.m. MISCELLANEOUS APPLICATIONS. 51 1 fc i 5i \^- i^ -1 _oq_ oi 5: ^v :V sdiM J 00 52 GRAPHICAL ALGEBRA. Example 2. A, B, and C run a race of 300 yards. A and C start from scratch, and A covers the distance in 40 seconds, heating G by 60 yards. B, with 12 yards^ start, beats A by 4: seconds. Supposing the rates of running in each case to he uniform, find graphically the relative positions of the runners when B passes the winning post. Find also by how many yards B is ahead of A luhen the latter has run threefourths of the course. In Fig. 25 let time be measured horizontally (0*5 inch to 10 seconds), and distance vertically (1 inch to 60 yards). O is the start- ing point for A and G; take OP equal to 0'2 inch, representing 12 yards, on the vertical axis ; then P is ^'s starting point. ^'s graph is drawn by joining O to the point which marks 40 seconds. From this point measure a vertical distance of 1 inch downwards to Q. Then since 1 inch represents 60 yards, Q is O's position when A is at the winning post, and OQ is C's graph. Along the time-axis take 1*8 inch to R, representing 36 seconds ; then PR is B's graph. Through R draw a vertical line to meet the graphs of A and G in S and T respectively. Then S and T mark the positions of A and G when B passes the winning post. By inspection RS and ST represent 30 and 54 yards respectively. Thus B is 30 yards ahead of A, and A is 54 yards ahead of G. Again, since A runs three-fourths of the course in 30 seconds, the difference of the corresponding ordinates of ^'s and B's graphs after 30 seconds will give the distance between A and B. By measure- ment we find VW = 0*45 inch, which represents 27 yards. The student is recommended to draw a figure for himself on a scale twice as large as that given in Fig. 25, 37. When a variable quantity ?/ is partly constant and partly proportional to a variable quantity x, the algebraical relation between jj? and y is of the form ?/ = a:c + b, where a and b are constant. The corresponding graph will therefore be a straight line ; and since a straight line is completely determined when the positions of two points are known, it follows that, in all problems w^hich can be illustrated by linear graphs, it is sufficient if the data furnish for each graph two independent pairs of simultaneous values of the variable quantities. Some easy examples of this kind have already been given on page 33 and in Examples VII. We shall now work out two more examples. 300r 270 ■ 240- 210 - 180- 150- 120- 90- eo- 30 - MISCELLANEOUS APPLICATIONS. 10 20 30 R 40 50 53 SeooLds J J 7 t t^ ^ 1 1 t 1 /is it -/- yf ^ T 4 t 7 t t T -Tl- J-^- 2 t 1 L-,W J -t 7-4- ' t 1 J^ L^l ' 4 t -L t-t ' -tt 7 2 ' ' M 7 11 -t t 1 - 1 t-T 7 X- il t% T 4 ' J tl ' -, t 7 7^ / L-t 7 -4 ' t tl 7 it T l-t 7 tt ' -T'J 11 ' ttl -4-1' ttl trt 111 IT tt' -,^1 1-P tl 4IT til tj i t Fig. 25. 54 GRAPHICAL ALGEBRA. Example L In a certain establishment the clerks are paid an initial salary for the first year^ and this is annually increased by a fixed bonus, the initial salary and the bonus beimj different in different departments. A receives £L30 in his 10^^ year, and £220 in his 19*''^. B, in another departmerit, receives £140 in his 5^^ year and £180 in his IS''^. Draw graphs to shew their salaries in different years. In what year do they receive equal salaries ? Also find in what year A earns the same salary as that received by B for his 2P* year. In Fig. 26 let each horizontal division represent 1 year ; and let the salaries be measured vertically, beginning at 130, with 1 division to represent £2. If the salary at the end of x years is denoted by £y, it is evident that in each case we have a relation of the form y = ax-{-b, where a and b are constant. Thus the variations of time and salary may be represented by linear graphs. Since no bonus is received for the first year, x = 9, when y = lSO, and x=\S, when y = 220. Thus the points P and Q are determined, and by joining them we have the graph for ^'s salary. Similarly the graph for B's salary is found by joining P' (4, 140) and Q' (12, 180). These lines have the same ordinate and abscissa at L, where a: = 16, 2/ = 200. Thus A and B have the same salary when each have served 16 years, that is in their 17^^ year. Again ^'s salary at the end of 20 years is given by the ordinate of M , which is the same as that of Q which represents ^'s salary after 18 years. Thus ^'s salary for his 19^^ year is equal to B's salary for his 21** year. Example 2. Two sums of money are put out at simple interest at different rates per cent. In the first case the Amounts at the end of 6 years and 15 years are £260 and £350 respectively. In the second case the Amounts for 5 years and 20 years are £330 and £420. Draw graphs from which the Amounts may be read off for any year, and find the year in which the Principal ivith accrued Interest will amount to the same in the two cases. Also from the graphs read off the value of each Principal. When a sum of money is at simple interest for any number of years, we have Amount = Principal + Interest, where 'Principal' is constant, and 'Interest' varies with the number of years. Hence the variations of Amount and Time may be represented by a linear graph in which x is taken to denote the number of years, and y the number of pounds in the corresponding Amount. Here as the diagram is inconveniently large we shall merely indicate the steps of the solution which is similar in detail to that of the last example. The student should draw his own diagram. MISCELLANEOUS APPLICATIONS. 55 / / ./ / 220 c H 1 H JM / J r ^ / / x / j / T' '~ U- rirjri t^ r_ "JL jf T 7 F 190 2_^ ,/ -f^ / / ll / |l , / / I5^n c 1/ / / / / / / / 17 n / / j ^l / ^ ~^/ ^ -^- ^ ^R^ <§r e "1 .^^y Jc "^Z- - ^ 1 _j 1 1 . / T 1 t / / \L / 140 t J T / / 1 130 J. Yedri 1 1 1 P 10 15 Fig, 26. 20 56 GRAPHICAL ALGEBRA. Measure time horizontally (1 inch to 10 years), and Amount" vertically (1 inch to £40) beginning at £260. The first graph is the line joining L (6, 260) and M (15, 350). The second graph is the line joining L' (5, 330) and M' (20, 420). In each of these lines the ordinate of any point gives the Amount for the number of years given by the corresponding abscissa. Again LM, L'M' intersect at a point P where 07 = 25, 2/ = 450. Thus each Principal with its Interest amounts to £450 in 25 years. When a; = there is no Interest ; thus the Principals will be obtained by reading off the values of the intercepts made by the two graphs on the y-axis. These are £200 and £300 respectively. Note. To obtain the result 2^=200 it will be necessary to continue the y-axis downwards sufficiently far to shew this ordinate. EXAMPLES VIII. 1. At noon A starts to walk at 6 miles an hour, and at 1.30 p.m. B follows on horseback at 8 miles an hour. When will B overtake ^? Also find (i) when A is 5 miles ahead of B ; (ii) when ^ is 3 miles behind B. [Take 1 inch horizontally to represent 1 hour, and 1 inch vertically to represent 10 miles.] 2. By measuring time along OX (1 inch for 1 hour) and distance along OY (1 inch for 10 miles) shew how to draw lines (i) from O to indicate distance travelled towards Y at 12 miles an hour ; (ii) from Y to indicate distance travelled towards O at 9 miles an hour. If these are the rates of two men who ride towards each other from two places 60 miles apart, starting at noon, find from the graphs when they are first 18 miles from each other. Also find (to the nearest minute) their time of meeting. 3. Two bicyclists ride to meet each other from two places 95 miles apart. A starts at 8 a.m. at 10 miles an hour, and B starts at 9.30 a.m. at 15 miles an hour. Find graphically when and where they meet, and at what times they are 37^ miles apart. MISCELLANEOUS LINEAR GRAPHS. 57 4. ^ and B start at the same time from London to Ellsworth, A walking 4 miles an hour, B riding 9 miles an hour. B reaches Ellsworth in 4 hours, and immediately rides back to London. After 2 hours' rest he starts again for Ellsworth at the same rate. How far from London will he overtake A, who has in the meantime rested 6^ hours ? 5. At what distance from London, and at what time, will a train which leaves London for Rugby at 2.33 p.m., and goes at the rate of 35 miles an hour, meet a train which leaves Rugby at 1.45 p.m. and goes at the rate of 25 miles an hour, the distance between London and Rugby being 80 miles ? Also find at what times the trains are 24 miles apart, and how far apart they are at 4.9 p.m. 6. A, B, and G set out to walk from Eath to Eristol at 5, 6, and 4 miles an hour respectively. C starts 3 minutes before, and B 7 minutes after A. Draw graphs to shew (I) when and where A overtakes G ; (ii) when and where B overtakes A ; (iii) C's position relative to the others after he has walked 45 minutes. [Take 1 inch horizontally to represent 10 minutes, and 1 inch to the mile vertically.] 7. X and Y are two towns 35 miles apart. At 8.30 p.m. A starts to walk from X to Y at 4 miles an hour ; after walking 8 miles he rests for half an hour and then completes his journey on horseback at 10 miles an hour. At 9.48 a.m. B starts to walk from Y to X at 3 miles an hour ; find when and where A and B meet. Also find at what times they are 6| miles apart. 8. A can beat B by 20 yards in 120, and B can beat G by 10 yards in 50. Supposing their rates of running to be uniform, find graphically how much start A can give G in 120 yards so as to run a dead heat with him. li A, B, and G start together, where are A and G when B has run 80 yards ? 9. A, B, and G run a race of 200 yards. A gives B a start of 8 yards, and G starts some seconds after A . A runs the distance in 25 seconds and beats G by 40 yards. B beats A by I second, and when he has been running 15 seconds, he is 48 yards ahead of G. Find graphically how many seconds G starts after A. Shew also from the graphs that if the three runners started level they would run a dead heat. [Take 1 inch to 40 yards, and 1 inch to 10 seconds.] 10. A cyclist has to ride 75 miles. He rides for a time at 9 miles an hour and then alters his speed to 15 miles an hour covering the distance in 7 hours. At what time did he change his speed ? 58 QRAPHICAL ALGEBRA. 11. A and B ride to meet each other from two towns X and Y which are 60 miles apart. A starts at 1 p.m., and B starts 36 minutes later. If they meet at 4 p.m., and A gets to Y at 6 p.m., find the time when B gets to X. Also find the times when they are 22 miles apart. When A is half-way between X and Y, where is ^ ? 12. The distance from London to Bristol is 1 19 miles ; if I were to set out at noon to cycle from London, riding 23 miles the first hour and decreasing my pace by 3 miles each successive hour, find graphically how long it would take me to reach Bristol. Also find approximately the time at which I should reach Faringdon, which is 48 miles from Bristol. 13. At 8 a.m. A begins a ride on a motor car at 20 miles an hour, and an hour and a half later B, starting from the same point, follows on his bicycle at 10 miles an hour. After riding 36 miles, A rests for 1 hr. 24 min. , then rides back at 9 miles an hour. Find graphi- cally when and where he meets B. Also find (i) at what time the riders were 21 miles apart, (ii) how far B will have ridden by the time A gets back to his starting point. 14. I row against a stream flowing 1^ miles an hour to a certain point, and then turn back, stopping two miles short of the place whence I originally started. If the whole time occupied in rowing is 2 hrs. 10 mins. and my uniform speed in still water is 4| miles an hour, find graphically how far upstream I went. [Take 1 '2 of an inch horizontally to represent 1 hour, and 1 inch to 2 miles vertically.] 15. One train leaves Bristol at 3 p.m. and reaches London at 6 p.m. ; a second train leaves London at 1.30 p.m. and arrives at Bristol at 6 p. m. ; if both trains are supposed to travel uniformly, at what time will they meet? Shew from a graph that the time does not depend upon the distance between London and Bristol. 16. At 7.40 a.m. the ordinary train starts from Norwich and reaches London at 11.40 a.m. ; the express starting from London at 9 a.m. arrives at Norwich at 11.40 a.m. : if both trains travel uniformly, find when they meet. Shew, as in Ex. 15, that the time is independent of the distance between London and Norwich, and verify this conclusion by solving an algebraical equation. 17. A boy starts f rom home and walks to school at the rate of 10 yards in 3 seconds, and is 20 seconds too soon. The next day he walks at the rate of 40 yards in 17 seconds, and is half a minute late. Find graphically the distance to the school, and shew that he would have been just in time if he had walked at the rate of 20 yards in 7 seconds. MISCELLANEOUS LINEAR GRAPHS. 59 18. The annual expenses of a Convalescent Home are partly constant and partly proportional to the number of inmates. The expenses were £384 for 12 patients and £432 for 16. Draw a graph to shew the expenses for any number of patients, and find from it the cost of maintaining 15. In a rival establishment the expenses were £375 for 5, and £445 for 15 patients. Find graphically for what number of patients the cost would be the same in the two cases. 19. A body is moving in a straight line with varying velocity. The velocity at any instant is made up of the constant velocity with which it was projected (measured in feet per second) diminished by a retardation of a constant number of feet per second in every second. After 4 seconds the velocity was 320, and after 13 seconds it was 140. Draw a graph to shew the A^elocity at any time while the body is in motion. A second body projected at the same time under similar conditions has a velocity of 450 after 5 seconds, and a velocity of 150 after 15 seconds. Shew graphically that they will both come to rest at the same time. Also find at what time the second body is moving 100 feet per second faster than the first, and determine from the graphs the velocity of projection in each case. 20. To provide for his two infant sons, a man left by his will two sums of money as separate investments at difi'erent rates of interest, on the condition that the principal sums with simple interest were to be paid over to his sons when the amounts were the same. After 5 years the first sum amounted to £451, and after 15 years to £533. After 10 years the second sum amounted to £432, and after 20 years to £544. Draw graphs from which the amounts may be read off for any year, and find after how many years the sons were entitled to receive their legacies. Also determine from the graphs what the original sums were at the father's death. 21. In a certain examination the highest and lowest marks gained in a Latin paper were 153 and 51. These have to be reduced so that the maximum (120) is given to the first candidate, and the minimum (30) to the lowest. This is done by reducing all the marks in a certain ratio, and then increasing or diminishing them all by the same number. In a Greek paper the highest and lowest marks were 161 and 56 ; after a similar adjustment these become 100 and 40 respectively. Draw graphs from which all the reduced marks may be read off, and find the marks which should be finally given to a candidate who scored 102 in Latin and 126 in Greek. Shew also that it is possible in one case for a candidate to receive equal marks in the two subjects both before and after reduction. What are the original and reduced marks in this case ? 60 GRAPHICAL ALGEBRA. Miscellaneous Graphs. 1. Plot the graphs of obtaining at least five points on each graph. Find the coordinates of the point where they meet. 2. Draw the graphs represented by y = 5-3x, y = ^{x-\-^) ; and find the coordinates of their point of intersection. 3. By finding the intercepts on the axes draw the graphs of (i) I5x + 20y = 6; (ii) l2x + 2ly=U, In (i) take 1 inch for unit, and in (ii) take six tenths of an inch as unit. In each case explain why the unit is convenient. 4. Solve y = l0x-i-8, 7x + y = 25 graphically. [Unit for x, one inch ; for y, one-tenth of an inch.] 5. From the graph of the expression l\x + Q, find its value when 07=1 '8. Also find the value of x which will make the ex- pression equal to 20. 6. With the same units as in Ex. 4 draw the graph of the 36 — 5x function — . From the graph find the value of the function when x=l'S ; also find for Avhat value of x the function becomes equal to 8. 7. Shew that the straight lines given by the equations 9y = 5x + 65, 5:r + 2^/ + 10 = 0, x + 3y=ll, meet in a point. Find its coordinates. 8. Draw the triangle whose sides are given by the equations . 3y-x=:9, x + 7y=U, 3x-^y = lS; and find the coordinates of its vertices. 9. Shew graphically that the values of x and y which satisfy the equations 5x = 2y-lS, 5y = 6-7x, also satisfy the equation x + y = 2. 10. Draw the graphs of (i) y = x'^, (ii) y = 8x'^. In (i) take 0*4" as unit for x, 0*2" as unit for y. In(ii) r X, O-r y. MISCELLANEOUS GRAPHS. 61 11. On the same scale as in Ex. 10. (ii) draw the graph of i/ = lQx^. Shew that it may also be simply deduced from the graph of Ex. 10. (ii). 12. Plot the graph of y = x'^, taking 1 inch as unit on both axes, and using the following values of x. -0-4, -0-3, -0-2, -01, 0, 01, 0-2, 03, 04. 13. Draw the graph oi x = y^^ from y = to 2/ =^ 5, and thence find the square roots of 7 and 3 '6. [Take 0*2" as unit for x^ \" as unit for 2/.] 14. Draw the graph of y — ^-\-x-x^ for values of x from - 2 to + 3, and from the figure obtain approximate values' for the roots of the equation 5 + ic - x^ = 0. [Take 1" as unit for x^ 0'2" as unit for y.] 15. Draw the graphs of (i) 5:i: + 6y = 60, (ii) %y-x^1^, (iii) 2x-y = 1', and shew that they represent three lines which meet in a point. 16. If 1 cwt. of coffee costs £9. 12s., draw a graph to give the price of any number of pounds. Read off the price (to the nearest p^nny) of 13 lbs., 21 lbs., 23 lbs. 17. If 60 eggs cost 4s., find graphically how many. can be bought for half-a-crown, and the cost of 26 eggs to the nearest penny. 18. If 1 cwt. of sugar costs £1. Qs. 8d., draw a graph to find the price of any number of pounds. Find the cost of 26 lbs. How many pounds can be bought for 4s. 10c?. ? 19. Solve the following equations graphically. (i) a:2 + y2^53, (ii) x^ + y^ = lOO, y-x = 5; x + y = l4:; (iii) ic2 + .v2=34, (iv) a;2 + 2/2^36, 2x^y=U ; 4:X + Sy=l2. [Approximate roots to be given to one place of decimals.] 20. Solve the equation S + Qx^x"^ graphically, and find the maximum value of the expression 3 + 6a: - x^. 21. A basket of 65 oranges is bought for 4s. 2d. Draw a graph to shew the price for any other number. How many could be bought for 3s. 4rf. ? Find the price (to the nearest penny) which must be paid for 36 and for 78 oranges respectively. E 62 GRAPHICAL ALGEBRA. 22. If the wages for a day's work of 8 hours are 4s. Qd. , draw a graph to shew the wages for any fraction of a day, and find (to the nearest penny) what ought to be paid to men who work 2^, 3|, 6^ hours respectively. How many hours' work might be expected for 2s. 10c?. ? [Take 1 inch to represent 1 hour, and one-tenth of an inch to represent 1 penny.] 23. Draw the graphs of x'^ and 3;r + l. By means of them find approximate values for the roots of a;^ - 3a? - 1 = 0. 24. If 24 men can reap a field of 29 acres in a given time, find roughly by means of a graph the number of acres which could be reaped in the same time by 15, 33, and 42 men respectively. 25. The highest marks gained in an examination were 136, and these are to be raised so that the maximum is 200. Shew how this may be done by means of a graph, and read off, to the nearest integer, the final marks of candidates who scored 61 and 49 respectively. 26. Draw a graph which will give the square roots of all numbers between 25 and 36, to three places of decimals. [Plot the graph of y = x^, beginning at the point (5, 25), with 10" and 0"5" as units for x and y respectively.] 27. I want a ready way of finding approximately 0*866 of any number up to 10. Justify the following construction. Join the origin to a point P whose coordinates are 10 and 8*66 (1 inch being taken as unit) ; then the ordinate of any point on OP is 0'866 of the corresponding abscissa. Read off" from the diagram, 0-866 of 3, 0-866 of 6*5, 0-866 of 4-8, and ^^^ of 5. 28. A starts from London at noon at 8 miles an hour ; two hours later B starts, riding at 12 miles an hour. Find graphically at what time and at what distance from London B overtakes A. At what times will A and 5 be 8 miles apart? If G rides after B, starting at 3 p.m. at 15 miles an hour, find from the graphs (i) the distances between A, B, and Cat 5 p.m. ; (ii) the time when C is 8 miles behind B. 29. If O and Y represent two towns 45 miles apart, and if A walks from Y to O at 6 miles an hour while B walks from O to Y at 4 miles an hour, both starting at noon, find graphically their time and place of meeting. Also read off from the graphs (i) the times when they are 15 miles apart ; (ii) B^s distance from Y at 6.15 p.m. MISCELLANEOUS GRAPHS. 63 30. At 8 a.m. A starts from P to ride to Q which is 48 miles distant. At the same time B sets out from Q to meet A. li A rides at 8 miles an hour, and rests half an hour at the end of every hour, while B walks uniformly at 4 miles an hour, find graphically (i) the time and place of meeting ; (ii) the distance between A and ^ at 11 a.m. ; (iii) at what time they are 14 miles apart. 31. The following table gives statistics of the population of a certain country, where P is the number of millions at the beginning of each of the years specified. Year 1830 1835 1840 1845 1850 1855 1860 P 20 22 24-5 28 31 36 41 Let t be the time in years from 1830. Plot the values of P vertically and those of t horizontally and shew the relation between P and ^ by a simple curve passing fairly evenly among the plotted points. Find what the population was at the beginning of the years 1847 and 1858. 32. The salary of a clerk is increased each year by a fixed sum. After 6 years' service his salary is raised to £128, and after 15 years to £200. Draw a graph from which his salary may be read off for any year, and determine from it (i) his initial salary, (ii) the salary he should receive for his 21st year. 33. Draw the graphs of y = a:^ and 2^/ = x + 3 on the same diagram. Deduce the roots of the equation 2x'^ - ic - 3 = 0. 34. Taking 1 inch as unit, plot the graph of y = x^-3x, taking the following values of x : 0, ±-2, ±-4, ±-6, ±'8, ±1, ±1-2, ±1-4, ±1-6, ±1-8, ±2. Find the turning points, and the value of the maximum or minimum ordinates between the limits given. 35. From the graph in Ex. 34 find to two places of decimals the roots of x^-3x = 0. 36. Solve the following pairs of equations graphically : (i) x + y = l5, (ii) x-y = S, (iii) ^^2 + ^2=13^ 072/ = 36; it:?/ = 18; xy = Q. 64 GRAPHICAL ALGEBRA. 37. An india-rubber cord was loaded with weights, and a measurement of its length was taken for each load as tabulated. Plot a graph to shew the relation between the length of the cord and the loads. Load in pounds - 10 12 17 21 23 25 Length in centimetres 36-4 37-7 40-5 43 44-3 45-4 What was* the length of the cord unloaded? 38. A manufacturer has priced a certain set of lathes ; the largest sells at £176, and the smallest at £40. He wishes to increase his prices so that the largest will sell at £200 and the smallest at £50. By means of a graph find an algebraical relation between the new price {P) and the old price {Q), and find to the nearest pound the new prices of lathes originally priced at £150, at £125. 10s., and at £78. 39. The mean temperature on the first day of each month, on an average of 50 years, had the following values : Jan. 1, 37°; Feb. 1, 38"; Mar. 1, 40°; April 1, 45° ; May 1, 50° June 1, 57° July 1, 62° Aug. 1, 62° Sept. 1, 59°; Oct. 1, 54°; Nov. 1, 46°; Dec. 1, 41°. Represent these variations by means of a smooth curve. [The difference of length of different months may be neglected.] 40. The price in pence of a standard Troy ounce of silver on January 1st in each of the ten years 1891-1900 was 45, 40, 36, 29, 30, 31, 28, 27, 27, 28. Draw a smooth curve shewing its value approximately at any time during these ten years. 41. A manufacturer wishes to stock a certain article in many sizes ; at present he has five sizes made at the prices given below : Length in inches 20 27 33 45 54 Price in shillings 11 14-5 20 35 48-5 Draw a graph to shew suitable prices for intermediate sizes, and find what the prices should be when the lengths are 30 in. and 46 in. LOGARITHMS. Antilogarithms. 69 Log. 1 2 3 4 5 6 7 8 9 12 3 4 5 6 7 8 9 •50 •51 •52 53 •54 3162 3236 33U 3388 3467 3170 3243 3319 3396 3475 3177 3251 3327 3404 3483 3184 3258 3334 3412 3491 3192 3266 3342 3420 3499 3199 3273 3350 3428 3508 3206 3281 3357 8436 3516 3214 3289 3365 3443 3524 3221 3296 3373 8451 3582 3228 3304 3381 3459 3540 1 1 2 12 2 12 2 12 2 1 2 2 3 4 4 3 4 5 3 4 5 3 4 5 3 4 5 5 6 7 5 6 7 5 6 7 6 6 7 6 6 7 •55 •56 •57 •58 •59 3548 3631 3715 3802 3890 3556 3639 3724 3811 3899 3565 3648 3733 3819 3908 3573 3656 3741 3828 3917 3581 3664 3750 3837 3926 3589 3673 3758 3846 3936 3597 •3681 3'767 3855 3945 3606 3690 3776 '3864 3954 8614 3698 3784 3878 ,3963 8622 8707 3793 3882 8972 1 2 2 12 3 12 3 12 3 12 3 3 4 5 3 4 5 3 4 5 4 4 5 4 5 5 6 7 7 6 7 8 6 7 8 6 7 8 6 7 8 •60 •61 •62 •63 •64 3981 4074 4169 4266 4365 3990 4083 4178 4276 4375 3999 4093 4188 4285 4385 4009 4102 4198 4295 4395 4018 4111 4207 4305 4406 4027 4121 4217 4315 4416 4036 4130 4227 4325 4426 4046 4140 4236 4335 4436 4055 4150 4246 4345 4446 4064 4159 4256 4355 4457 1 2 3 12 3 1 2 3 12 3 1 2 3 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 6 7 8 7 8 9 7 8 9 7 8 9 7 8 9 •65 •66 •67 •68 •69 4467 4571 4677 4786 4898 4477 4581 4688 4797 4909 4487 4592 4699 4808 4920 4498 4G03 4710 4819 4932 4508 4613 4721 4831 4943 4519 4624 4732- 4842 4955 4529 4634 4742 4853 4966 4539 4645 4753 4864 4977 4550 4656 4764 4875 4989 4560 4667 4775 4887 5000 1 2 3 1 2 3 12 3 1 2 3 1 2 3 4 5 6 4 5 6 4 5 7 4 6 7 5 6 7 7 8 9 7 9 10 8 9 10 8 9 10 8 9 10 70 •71 •72 •73 •74 5012 5129 5248 5370 5495 5023 5140 5260 5383 5508 5035 5152 5272 5395 5521 5047 5164 5284 5408 5534 5058 5176 5297 5420 5546 5070- 5188 5309 5433 5559 5082 5200 5321 5445 5572 5093 5212 5333 5458 5585 5105 5224 5346 5470 5598 5117 5236 5358 5483 5610 1 2 4 12 4 12 4 1 3 4 13 4 5 6 7 5 6 7 5 6 7 5 6 8 5 6 8 8 9 11 8 10 11 9 10 11 9 10 11 9 10 12 •75 •76 •77 •78 •79 5623 5754 5888 6026 6166 5636 5768 5902 6039 6180 5649 5781 5916 6053 6194 5662 5794 5929 6067 6209 5675 5808 5943 6081 6223 5689 5821 5957 6095 6237 5702 5834 5970 6109 6252 5715 5848 5984 6124 6266 5728 5861 5998 6188 6281 5741 5875 6012 6152 6295 13 4 1 3 4 1 3 4 13 4 1 3 4 5 7 8 5 7 8 5 7 8 6 7 8 6 7 9 9 10 12 9 11 12 10 11 12 10 11 18 10 11 13 •80 •81 •82 •83 •84 6310 6457 6607 6761 6918 6324 6471 6622 6776 6934 6339 6486 6637 6792 6950 6353 6501 6653 6808 6966 6368 6516 6668 6823 6982 6383 6531 6683 6839 6998 6397 6546 6699 6855 7015 6412 6561 6714 6871 7031 6427 6577 6730 6887 7047 6442 6592 6745 6902 7063 1 3 4 2 3 5 2 3 5 2 3 5 2 3 5 6 7 9 6 8 9 6 8 9 6 8 9 6 8 10 10 12 13 11 12 14 11 12 14 11 13 14 11 13 15 •85 •86 •87 •88 •89 7079 7244 7413 7586 7762 7096 7261 7430 7603 7780 7112 7278 7447 7621 7798 7129 7295 7464 7638 7816 7145 7311 7482 7656 7834 7161 7828 7499 7674 7852 7178 7345 7516 7691 7870 7194 7362 7534 7709 7889 7211 7379 7551 7727 7907 7228 7396 7568 7745 7925 2 3 5 2 3 5 2 3 5 2 4 5 2 4 5 7 8 10 7 8 10 7 9 10 7 9 11 7 9 11 12 13 15 12 13 15 12 14 16 12 14 16 13 14 16 •90 •91 •92 •93 •94 7943 8128 8318 8511 8710 7962 8147 8337 8531 8730 7980 8166 8356 8551 8750 7998 8185 8375 8570 8770 8017 8204 8395 8590 8790 8035 8222 8414 8610 8810 8054 8241 8433 8630 8831 8072 8260 8453 8650 8851 8091 8279 8472 8670 8872 8110 8299 8492 8690 8892 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 7 9 11 8 9 11 8 10 12 8 10 12 8 10 12 13 15 17 13 15 17 14 15 17 14 16 18 14 16 18 •95 •96 •97 •98 •99 8913 9120 9333 9550 9772 8933 9141 9354 9572 9795 . 8954 9162 9376 9594 9817 8974 9183 9397 9616 9840 8995 9204 9419 9638 9863 9016 9226 9441 9661 9886 9036 9247 9462 9683 9908 9057 9268 9484 9705 9931 9078 9290 9506 9727 9954 9099 9811 9528 9750 9977 2 4 6 2 4 6 2 4 7 2 4 7 2 5 7 8 10 12 8 11 13 9 11 18 9 11 18 9 11 14 15 17 19 15 17 19 15 17 20 16 18 20 16 18 20 70 GRAPHICAL ALGEBRA. A Table of Square Boots and Cube Roots of Numbers from 1 to 150. Square No. Cube Square No. Cube Square No. Cube Root. Root. Root. Root. Root. Root. 1-000 1 1-000 7-141 51 3-708 10-050 101 4-657 1-414 2 1-260 7-211 52 3-73? 10-100 102 4-672 1-732 3 1-442 7-280 53 3-756 10-149 103 4-688 2-000 4 1-587 7-348 54 3-780 10-198 104 4-703 2-236 5 1-710 7-416 55 3-803 10-247 105 4-718 2-449 6 1-817 7-483 56 3-826 10-296 106 4-733 2-646 7 1-913 7-550 57 3-849 10-344 107 4-747 2-828 8 2-000 7-616 58 3-871 10-392 108 4-762 3-000 9 2-080 7-681 59 3-893 10-440 109 4-777 3-162 10 2-154 7-746 60 3-915 10-488 110 4-791 3-317 11 2-224 7-810 61 3-936 10-536 111 4-806 3-464 12 2-289 7-874 62 3-958 10-583 112 4-820 3-606 13 2-351 7-937 63 3-980 10-630 113 4-835 3-742 14 2-410 8-000 64 4-000 10-677 114 4-849 3-873 15 2-466 8-062 65 4-021 10-724 115 4-863 4-000 16 2-520 8-124 66 4-041 10-770 116 4-877 4-123 17 2-571 8-185 67 4-062 10-817 117 4-891 4-243 18 2-621 8-246 68 4-082 10-863 118 4-905 4-359 19 2-668 8-307 69 4-102 10-909 119 4-919 4-472 20 2-714 8-367 70 4-121 10-954 120 4-932 4-583 21 2-759 8-426 71 4-141 11-000 121 4-946 4-690 22 2-802 8-485 72 4-160 11-045 122 4-960 4-796 23 2-844 8-544 73 4-180 11-091 123 4-973 4-899 24 2-884 8-602 74 4-198 11-136 124 4-987 5-000 25 2-924 8-660 75 4-217 11-180 125 5-000 5-099 26 2-962 8-718 76 4-236 11-225 126 5-013 5-196 27 3-000 8-775 77 4-254 11-269 127 5-027 5-292 28 3-037 8-832 78 4-273 11-314 128 5-040 5-385 29 3-072 8-888 79 4-291 11-358 129 5-053 5-477 30 3107 8-944 80 4-309 11-402 130 5-066 5-568 31 3-141 9-000 81 4-327 11-446 131 5-079 5-657 32 3-175 9-055 82 4-344 11-489 132 5-092 5-745 33 3-208 9-110 83 4-362 11-533 133 5-104 5-831 34 3-240 9-165 84 4-380 11-576 134 5-117 5-916 35 3-271 9-220 85 4-397 11-619 135 5-130 6-000 36 3-302 9-274 86 4-414 11-662 136 5-143 6-083 37 3-332 9-327 87 4-431 11-705 137 5-155 6-164 38 3-362 9-381 88 4-448 11-747 138 5-168 6-245 39 3-391 9-434 89 4-465 11-790 139 5-180 6-325 40 3-420 9-487 90 4-481 11-832 140 5-192 6-403 41 3-448 9-639 91 4-498 11-874 141 5-205 6-481 42 3-476 9-592 92 4-514 11-916 142 5-217 6-557 43 3-503 9-644 93 4.531 11-958 143 5-229 6-633 44 3-530 9-095 94 4-547 12-000 144 5-241 6-708 45 3-557 9-747 95 4-563 12-042 145 5-254 6-782 46 3-583 9-798 96 4-579 12-083 146 5-266 6-856 47 3-609 9-849 97 4-595 12-124 147 5-278 6-928 48 3-634 9-899 98 4610 12-166 148 5-290 7-000 49 3-659 9-950 99 4-626 12-207 149 5-301 7-071 50 3-684 10-000 100 4-642 12-247 150 5-313 ANSWERS. 71 ANSWERS. I. Page 4. 7. 36. 8. 32. 9. 25. 11. l-2sq. cm. 12. y = Sx. Any point whose ordinate is equal to three times its 14. The lines are x = 5, y = S. The point (5, 8). 15. A circle of radius 13 whose centre is at the origin. II. Page 7. 21. 32 units of area. 22. 1 sq. in. 23. 72 units of area. 24. (>'64 sq. cm. III. Page 10. 1. x=l,y = 5. 2. x = 2, i/=\0. 3. x = S, y=\2. 4. x = S,y=-2. 5. x = 4:,i/ = 2. 6. x = G,y = H. 7. x=-2,y = 4. 8. x = 0,y=-S. 9. x=-3,y = 0. 10. At the point (0, 21 ). 11. Sx-{-4y = 7. IV. Page 15. 1. y = x. 2. (0, 0), (-4, 2). 6. (2,1). 6. (i) 1 -46, - 5-46 ; (ii) 3*24, - 1 24 ; (iii) 3-32, 68. 7. -5; 7. ^' ~l' ^'^^' -0-79; 462, -1*62. 9. x = S, or 6; y = 6, or 8. 10. The straight line 3x + 4:y = 25 touches the circle x'^ + y'^ = 25 at the point (3, 4). V. Page 24. 3. Each axis is an asymptote to the curve, which approaches the axis of y much less rapidly than it does the axis of x. 7. x = 2,^;y = 5,3. 8. a; = 3, - 3 ; 2/ = 2, -2. 9. x = 2;y=-l. 31. -1,1,2. 32. - 2, 4-41, 159. VI. Page 30. 1. 0-52, 2-9, 11-6 ; 275, 23, 31. 3. 2080, 2-140. 4. 2-4. 5. -2, 4; -9. 9. -5andl. 10. -2,1,4. 12. 26-9, 38, 3-58. 13. 0*477, 225, 350, 1538. 14. 3 in. from the point of suspension. 72 GRAPHICAL ALGEBRA. 15. 22 lbs., 16i lbs., 14| lbs., 13^ lbs., 11 lbs., 9^ lbs., 8 lbs., 7J lbs. The curve is a rectangular hyperbola whose equation is 17. 0-602. 53°. 18. 0906. 0-707. 19. a; = ?i7r + ^. 20. The range varies from 342 yards, when ^ = 10°, to 985 yards when ^ =40°. It reaches its maximum of 1000 yards when A =45°, and is again equal to 985 yards, when A = 50°. 21. 5-9°, 7-7°. VII. Page 40. 2. (i) 64-1 grains ; (ii) 0-2. 3. 39*3; 91*6; 2/ = 0-393a;. 4. 3-85 in. ; 17*6 in. 5. 54-5°F. 86-9°F. F = 324-^C. 5 6. 2/= 100 + :^; £350; 4250. 7. 45 96; 39 40. 8. £2. 12,s'. ; £3. Ss. 9. 81 in. ; 24 375 oz. 10. (i) £320; (ii) £580. 11. y = ^x-10. 112; 168; 78. 12. 5 ft. per sec; 5|secs.; v = 5 + 4^. 13. 2*49 sq. ft. 14. (i)52ft.; (ii) 160 ft. 15. max. height = 64 ft.; 4 sees. 16. P = 6 6^-14-4; 24. 17. 26.s.; 36.9. 6^. 18. 93-5° E. 20. 2/ = 0-21a;+l-37. 21. i/ = 249-8-250a;. 22. i/ = 0-4a; + l-6 ; 92; 3. 23. a = 45-7, ?) = 118. Error = 8-43 in defect. 24. 8-6; P = 014Pr+0'2; 225 lbs. 25. a = i;6 = 3. 2; 12. 26. a = 3, 6 = 2. 7 ; 425. 27. 71 = 3, c = 27 X 10^. 28. 9i=l-5, c = 79500. 17 29. 1 '3 per cent, in defect. 31. '^ = t^- 32. 16 knots. 33. H.P. =6 9. 23 lbs. 34. 1515 knots; 2420 H. P. 35. P = 200, r = 3; 14 years. 36. £22. 10.-?.; £28. (i) 230; (ii) 350. 1 100 A'= 496? + 8140 ; 550i? = 52G^-2255; 20P = (?-230. 37. ^=0-84 W^+ 10-2; i? = 2-32Pr- 305 ; P=l-48 W^- 407. (i) 40 ; (ii) 28. 38. 7^=36 •31/^15; V=l21\h}\ 39. 71 = 0-86, c = 207. 40. a=l-78, 6 = 31-5. 41. i) = 15-85/ii8: 669 tons. ANSWERS. 73 VIII. Page 56. 1. 6 p.m. ; (i) 3.30 p.m. ; (ii) 7.30 p.m. 2. (i) 2 p.m. ; 2.52 p.m. 3. 47 mi. from -4's starting place at 12.42 p.m. 11.12 a.m. and 2.12 p.m. 4. 27 mi. 5. 35 mi. from London at 3.33 p.m. 3.9 p.m. ; 3.57 p.m. ; 36 mi. 6. (i) 15 mi. after (7's start, 1 mi. from Bath ; (ii) 45 3^ mi ; (iii) half a mile behind A and B. 7. 9 mi. from Y at 12.48 p.m. 12.18 p.m. and 1.18 p.m. 8. 40 yds. A 16 yds. ahead, C 16 yds. behind. 9. 5 sees. 10. 5 hours from the start. 11. 7.36 p.m. ; 3 p.m. and 5 p.m. ; 19 mi. from Y. 12. 7 hours. 4.47 p.m. 13. 12.12 p.m. (i) 11 a.m. ; (ii) 57 mi. 14. 5 mi. 15. 4.12 p.m. 16. 10.4 a.m. 17. 400 yds. 18. £420. 20 for £480. 19. After 10 sees. 400 ft. per sec. ; 600 ft. per sec. 20. 30 years. £410; £320. 21. 75 in Latin ; 80 in Greek. 74 and 50. Miscellaneous Graphs. Page 60. 1. (2,-3). 2. (1,2). 4. x=l,y=lS. 5. 26 ; 1 '28 (approx. ). 6. 9 ; 2-4. 7. ( - 4, 5). 8. (3, 4), (4, 1), ( - 3, 2). 13. 2-65, 1-91. 14. 2-79, -1-79. 15. The pt. (6, 5). 16. 22s. Sd. ; 36s., 39s. 5d. 17. 37 ; Is. 9d. 18. 6s. 2d. ; 203. 19. (i) x = 2, or -7; y = 7, or -2. (ii) x = 8, or 6; y = 6, or 8. (iii) a; = a, or -5*8; y = 5, or -0-6. (iv) x = 5'2, or -1*3; y= -2*9, or 5-7. 20. 6-46, -0-46; 12. 21. 52; 2s. 4d., 5s. 22. Is. 5d., 2s., .3s. Sd. ; 5 hrs. 23. 3.30, -030. 24. 18, 40, 51. 25. 90, 72. 27. 2*60, 563, 416, 5-77. 28. 6 p.m., 48 mi. from London. At 4 and 8 p.m. (i) B 4 mi. behind A; (7 6 mi. behind B. (ii) 4.21 p.m. 29. 4.30 p.m., 18 mi. from O. (i) At 3 and 6 p.m. (ii) 20 mi. 30. (i) 1 p.m., 28 mi. from P; (ii) 20 mi. ; (iii) 11.30 a.m. 31. 29| and ,39 millions. 32. £80 ; £240. 33. 1-5, -1. 34. Max.ordinates( = 2)atthepoints(-l,2),(l, -2). 74 GRAPHICAL ALGEBRA. 35. -1-73, 0, 1-73. 36. (i) x=12, or 3\ (ii) x = 6, or -3\ (iii) x=2, 3, -3, -2\ y = S, or 12/ y = S, or -6j y = S, 2, -2, -3/ 37. 31'4cm. 38. P-lig + 6 ; £171 ; £144 ; £92. 41. 17s. ; 26s. 6d. 42. 4^ mi. per hr. 43. 1.30 p.m. ; 3 : 1. 44. 24 min. 45. At Northampton. 48-4 mi., 84*8 mi. 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