-110 the axes at distances from the origin — C+A, —C+B 1'espect_i:cly), yet in dealing by this method with descriptive propositions, _we are, 111 fact, eminently free from all metrical considerations. 7. It is worth while to illustrate this by the instance of the well-known theorem of the radical centre of three circles. The theorem is that, given any three circles A, I}, C (fig. 8), the common chords aa', , 77’ of the three pairs of circles meet in a point. The geometrical proof is metrical throughout :— Take 0 the point of inteisection of cm’, BB’, and joining this with 7', suppose that 7'0 does not _ _ pass through 7, but that it meets the circles A, B in two distinct points 7,, 7, respectively. 'e have then the known metrical pro- perty of intersecting chords of a circle; viz. , in circle C where aa’, BB’ are chords meeting at a point 0, Oa.Oa'=OB.OB', where, as well as in what immediate follows Oa, &c., denote, of course, lengths or (7z'stm2t.'€s. Similarly in circle A, I ‘ OB.OB =-O71.U7', and in circle 13, 0a. 01:.’ = 07: . 07'. Consequently O7, . O7’= 07,. 07’, that is, _O7,=O72, or the points 71 and 7, coincide; that is, they each coincide with 7. ‘Ye contrast this with the analytical method :— Here it only requires to be known that an equation A9: + By + C = 0 represents a line, and an equation :r:'"’ + 3/9 + Ax + By + C = 0 represents a circle. A, B, C have, in the two cases respectively, metrical signi- fications; but these we are not concerned with. Using S to denote the function 99+ 1/'3 +A:c+ B3/+C, the equation of a circle is S=O, where S stands for its value ; more briefly, we say the equation is S, =x9 + ,1/'3 + Arc + By + 0, =0. Let the equation of any other circle be S’, =f+3/"’ +A’;v+ B'y+ C'= 0; the equation S — S’==O is a linear equation (S — S’ is in fact = (A — A’) + (B — B’) 3/ + C — C’), andit thus represents a line ; this equation is satisfied by the co- ordinates of each of the points of intersection of the two circles (for at each of these points S = 0 and S'= 0, therefore also S — S’=0); hence the equation S — S'=0 is that of the line joining the two points of intersection of the two circles, or say it is the equation of the common chord of the two circles. Considering then a third eircle S”, -=x'-’+3,/2 + A”~.v+ B";/+ C”=0, the equations of the coni- mon chords are S—S'=O, S—S”=0, S’ —S” =0 (each of these a linear equation) ; at the intersection of the first and second of these lines S—=S’ and S=S", therefore also S’=S", or the equation of the third line is satisfied by the coordinates of the point in question ; that is, the three chords intersect in a point 0, the co- ordinates of which are determined by the equations S=S’=S”. It further appears that if the two circles S =0, S = 0' do not intersect in any real points, they must be regarded as intersecting in two imaginary points, such that the line joining them is the real line represented by the equation S — S’ = 0 ; or that two circles, whether their intersections be real or imaginary, have always a real common chord (or radical axis), and that for any three circles the coin- moii chords intersect in a point (of course real) which is the radical centre. And by this very theorcni, given two circles with imaginary intersections, we can, by drawing circles which meet each of them in real points, construct the radical axis of the first-mentioned two circles. 8. The principle employed in showing that the equation of the common chord of two circles is S — S’ = 0 is one of very extensive application, and some more illustrations of it may be given. Suppose S=O, S’=0 are lines (that is, let S, S’ now denote linear functions .:c+ By+ C, A’;c t I3’;/+ C’), then S — IcS’=0 (I; an arbitrary constant) is the equation of any line passing through the point of intersection of the two given lines. Such a line may be G-EOMETRY [rL..'E Jt.TAI.Y'l‘IC‘AL. made to pass through any given point, say the point (J0, yo) ; 2'.c., if S,,, S’0 are what S, S’ respectively become on writing for (,,_-, 3/) the values (.ro, yo), then the value of I: is l:=S.,—:-S’(,. The equation in fact is SS'o—S,,S'==O ; and starting from this equation we at one.- verify it apostcriori ; the equation is a linear equation satisfied by the values of (.15, 3/) which make S=O, S’-0 ; ainl satisfied also by the values (1-0, yo); and it is thus the equation of the line in question. If, as before, S=O, S'=0 represent circles, then (I; bt-in_r_; arbi. trary) S —k Q’=0 is the equation of any circle passing through the two points of intersection of the two circles ; and to make this 1-:l.S through a given point (.2-0, yo) we have again l.'—=S,,+S’0. in the particular case k=1, the circle. becomes the coimnoii chord (more accurately, it becomes the common chord together with the line infinity, but this is a question which is not here gone into). If S denote the general quadrie function, S=aJ:2+27L1‘y+b_I/9+2/'_I/+2g;c+c, =(r(, I», c, f, 3/, 7w_.(.7', 3/, 1)”, then the equation S=0 represents a conic ; assinnin_-.5 this, then. if S’=O represents another conic, the equation S —I.'."_ -o i'ep1-t-sent»; any conic through the four points of intersection of the two mnics. Returning to the equation Ax+l’»y+C=O of a line, if this 1).’i..s through two given points ((131, y,), (any, 3/_,), then we nmst hm-.- Ax, + By, + C = 0, A;r2 + By, + C = 0, equations which dctcrininc. the ratios A : B :C, and it thus appears that the equation of the line through the two given points is =0; all/i ‘ 3/2) ‘ 3/(.-1'1 " 9'2) + ‘T17 :2 ‘ 9723/1 = 0 i or what is the same thing- 9', 3/, 1 9'1 a 3/1 a 1 ‘T2 3 2/2! 9 1 9. The object still being to illustrate the mode of working with coordinates, we consider the theor.-m of the polar of a point in regard to a circle. Given a circle and a point (J (lig. 9), we draw through 0 any two lines 33' meeting the circle in the points A, A’ and l}, 1' re- B spectively, and then taking 0 Q as the intersection of A A the lines AB’ and AT», the theorem is that the locus of the point Q is a Fit, 9 right line depending only °' ' upon 0 and the circle, but independent of the particular lines 0AA’ and O1} ". Taking O as the origin, and for the axes any two lines through 1 P at right angles to each other, the equation of the circle will be
- 10‘-’+,7/'“'+2.t::'+‘2l‘.y+('=0;
and if the equation of the line O.--' is taken to be y==m::.-, then the points A, A’ are found as the intersections of the straight lin- with the circle ; or to determine .v we have
- L"’(1 + 721”) + 2.'r:(. + Bm) + C= 0 .
If (.13, y,) are the coordinates of A, and (.-*2, 3/._.) of .1’, then tl.-_ roots of this equation are 9'1, 9'2, whence easily l_+_1_= _ 2 i+l3m . .7", .73., C- And similarly, if the equation of the line 0]‘. 1' is taken to be y=m':r:, and the coordinates of B, B’ to be (.13, 3/3) and (.'r',, 3/4) respectively, then We have then -Ti?/1 ‘ 3/4) " ?/("'1 ‘ 3'4) +-Ti.’/4 " ‘T-I2/1 = 0 v 9?(?/2 ‘ 3/3) “ J(-"'2' i"3)+7'-.-3/:1‘ ‘’-'33/-1 ‘ 0 » as the equations of the _lines A_l’;: and it'll rcspecti'I-ly_: for the first of these equations, being satished if we write ,lllt'l‘('lll_ (.:_-I, _1/1) or (g',, 7/4) for (.r, y), is the equation of the line .B, and snnilarly the second equation is that of the line _~'l}. ltcdut-ing by inc-ans: of the relations yl — ma‘, = 0, y., — 711.7‘, = -0, 3/3 — ')n'.r3 = U, 3/, — m'.':'_,— H, the two equations become a:(-ma‘, — m’:r_,) — y(.z'1 — 9'4) + (M: — 7n)fc,n', : O,
- r(m:v._, — m’.r3) — y(:r._. — 9'3) + (m — 7I1):c,-‘I73 —— C,
and if we divide the ll1I‘.‘l3t of t.h]ese_equations by 71117114, and the second by m2'm3, and then an , we 0 itain 511_._L1l_§l-1-_1_ll xtm "M .c,+.7‘, i 3! a.'3+:c:'(a:,+:r., l +2/n —- 2m= ,
or, what is the same thing,
