412 (i 11 O )1 inclination a to the axis of .1: And, inoreover, if, using this eq1ia- tioii (=p), we write .c=a.+p cos a, 3/-=b+p sin a, then we have expressions for the coordinates .1‘, 3/ of a point of this line, in terms of the variable paranietei' p. Again, take the point '1' to be fixed, but consider the point 1’ as moving in the line T1’ at right angles to QT. if instead of 51 we take 1) for the distance QT, then the equation f,=f cos a'+ 1; sin a’ will be (.z:—a) cosa’+(_2/— b) sin a'=p; that is, this will be the equation of a line such that. its perpendi- cular distance from the point (a, b) is =2), and that the inclination of this distance to the axis of J: is = a’. From either form it appears that the equation of a line is, in fact, a linear equation of the form A.z'+ 133/ + C = 0. It is important to notice that, starting from this equation, we can determine conversely the a. but not the (u, I3) of the form of equation which contains these quantities ; and in like manner the a’ but not the (a, la) or 1) of the other form of equation. The reason is obvious. In each case (u, (3) denote the coordinates of a point, fixed in- deed, but which is in the first form any point of the line, and iii the second form any point whatever. Thus, in the second form the point from which the perpendicular is let fall may be the origin. Here (/1, b)=(O, O), and the equation is :5 cos a'+_7/ siii a'—p=0. Comparing this with A.’I:+ B3/+ C = 0, we have the values of cos a.', sin a’, and 1;. 13. The equation p‘-’ = («s - as + (.3 - or is an expression for t-he squared distance of the two points (It, I3) and (.r, 3/). Taking as before the point Q, coordi- nates (a, b), as a fixed point, and writing c in the place of p, the equation <~c- «>‘-’+ <.«/— or-’=c=’ expresses that the point (.c, 3/) is always at a given distance c from the given point (a, II) ; viz. , this is the equation of a circle, having (:1, 1;) for the coordinates of its centre, and c for its radius. The equation is of the form .I'2 + 3/2 + 2.J; + 2133/ + U = 0, and here, the number of constants being the same, we can identify the two equations ; we find a — A, I): — 1-}, c'3=A‘-’+l}'-’—C, or the last equation is that of a circle having — A, -1} for the coordinates of its centre, and N/A"-’ + B2 — C for its radius. 14. Drawing (fig. 11) Q3/, at right angles at Qx, , and taking Q.r:,, Q3/1 as a new set of rectangular axes, if instead of £1, 1;, we write .r,, 3/1, we have ail, 3/1 as the new coordi- nates of the point P; and writing also a. in place of o.’ (a. now denoting the inclination of the axes (3.1-, and ().c), we have the forniuhe for transformation between two sets of rectangular axes. These are .r —a -= :73, cos a—_i/, sin a, 3/—b == 2:, sin a+3/1 cosa, and 9', =- 3/1 each set being obviously at once deducible from the other one. In these formula: ((1, 6) are the .7c3/—coordinates of the new origin Q1, and a. is the inclination of Qx, to Ox. It is to be noticed that Q.c,, Q3/1 are so placed that, by moving 0 to Q, and then turning the axes 0.:-1, 03/, round (2 (through an angle a measured in the sense O.r to ()3/), the original axes O./3, 03/ will come to coincide with 4.31,, Q3/, respectively. This could not have been done if Q3/1 had been drawn (at right angles always to Q13) in the reverse direction, we should then have had in the formulae —_2/, instead of 3/]. would be thus obtained are of an essentially distinct forni : the analytical test is that in the formulae as written (ar:—a) cos a+(_2/—b) sin a. = -(.z:—a) sin a+(_i/—b) cos a, E '1' i: 1' down we can, by giving to 0. a proper value (in fact a=0), make the (.:'—a) and (3/—-1;) equal to :1‘, and 3/, respectively; in the other system we could only make them equal to .:'1,— 3/1, or ~.r,, 3/, respectively. But for the very reason that the second system can be so easily derived from the first-, it is proper to attend exclusively to the first systeiii,—-tliat is, always to take the new axes so that the two sets admit of being brought into coincidence. In the foregoing system of two pairs of equations, the first pair give the original coordinates .r, 3/ in terms of the. new coordinates 2-,, 3/, ; the second pair the new eo- 0l‘Lli1)t1tcS.'l'1, 3/, in terms of the original coordinates .1‘, 3/. The forinulze involve (:1, b), the ori_-_,-'ina1 coordinates of the new origin; it would be easy instead of these to iiitro- duce (a,, bl), the new coordinates of the origin. 'i-itin-_{ (/1, I») = (O, O), we have, of course, the formul-.e for trans- formation between two sets of rectangular axes /uu-z'n_«/ //u .s-mm: or/'3/in, and it is as well to write the fU1‘l111ll;L‘ in this more simple form; the subsequent transformation to a new origin, but with axes parallel to the original axes, can then be effected without any difficulty. 15. All questions in regard to the line may be solved by means of one or other of the foregoing forins— A.r+li3/+C=O, [l’L.-‘.'E .i.'_u.vTicAi.. 3/=A.ir+B, w__-_“ = 2/-5 osa siiia (.2: - a) cos a’+(3/- (2) sin a’ —p=O ; or it may be by a comparison of these difl'crent foims : thus, using the first form, it has been already shown that the equation of the line through two given points (.r,, 3/1), (*2: .’/2) 15 nstyi ~ 2/2) — 2/(-vi - 9'2) +9—'12/-_- - 1'-.»y1 = 0» or, as this may be written, 3/ -7 . . 3/’ 2/1: ‘,2’ fit" _a’.l)' .72 — .4 1 A particular case is the equation a. _ - =1 (1. + I) ’ representing the line through the points (:1, O) and (O, 1;), or, what is the same thing, the. line meeting the axes of .r and 3/ at the distances from the origin It and I; respectively. It . . . A may be noticed that, in the form A.r+ 133/ + U : 0 , -1, denotes the tangent of the inclination to the axis of .r, or we may say that B + ./A‘~’ + 15'-' and — A + N/A3 +133 lenotc respectively the cosine and the sine of the inclination to the axis of 9:. A better form is this : A + ,/A'-' + 1F and I}—:- ,/112+ B‘-’ denote respectively the cosine and the sine of inclination to the axis of ./J of the perpendicular upon the line. So of course, in regard to the form 3/ =A.r+ I}, A is here the tangent of the inclination to the axis of .1‘; 1+ ,JA‘-’-+- Tand A+ ,/‘X311 are the cosine and sine of this inclination, &c. It thus appears that the condition in order that the lines Ar +1}3/ + C = Oaiid A'.r +]'i'!/— 0' = 0 may meet at right angles is AA'+ 131$’: 0 ; so when the equations are 3/ = A.v+ T», 3/ = A'.r_' + 13', the condition i< AA' +1 =0, or say the value of A’ is: -1 + A. The perpendicular distance of the 1)0ll1i£ /I) fi'oiii the line A.:-. + By + C = 0 is (An + 1373+ C) + ,/A’ +13”. In all the formuhc involving ,/Ait-—J3"_0I‘ ,/.’+ 1, the lmlicill slionld be written with the sign =h, which is essentially indetermina.te: the like iiidetcrmiiiatciicss of sign presents itself in the expression for the distance of two points = :1: J(1r—(1)2+(!/ ~ 1,)‘-’; if, as before, the points are The new formulae which : Q, l’, and the indefinite line through these is z’Q1’z, then it is the same thing whether we measure off from Q along - n I I ' ' V
this line, considered as drawn from 7. towards :1, a 1)0S1t1V0