414 3’ oblique axes) the equation of the liyperbola takes the form x3/=c , and in particular, if in this equation the V -C 0 (C Fe. 14. Fig. 13. '° 7/ ' I O -c Fig. 15. axes are at right angles, then the equation represents the rectangular liyperbola referred to its asymptotes as axes. Tangent, Normal, Circle and Ra(lz'u.s of C’ urvafurc, (ST. 20. There is great convenience in using the language and notation of the infinitesimal analysis ; thus we consider on a curve a point with coordinates (.1-, 3/), and a consecu- tive point the coordinates of which are (Jc+cl.r, 3/+d_2/), or again a second consecutive point with coordinates (ac + (1.1: +?§d2.17, _1/ + d 3/ + §d2_I/), &c. ; and in the final results the ratios of the infinitesimals must be replaced by differ- ential coeflicients in the proper manner; thus, if av, 3/ are considered as given functions of a parameter 6, then (lx, rl_z/ dy ’ d0 being really material) they may in the result be replaced by ‘L1 ‘L ' 1149 ’ (I0 ' of the curve is given in the form 3/= gb(.r) ; 6 is here = 9:, and the increments (l.'c, (l_I/ are in the result to be replaced have in fact the values gclél (I6, and (only the ratio This includes the case where the equation I . . . . . . by 1, :15: . So also with the iiifiiiitesiinals of the higher orders d‘-’.r, &c. 21. The tangent at the point (.r, 3/) is the line through this point and the consecutive point (J3 + cl.r, 3/ + 173/); hence, taking 5, 1; as current coordinates, the equation is €‘_“ = 71: ?/__ dcc dy ' an equation which is satisfied on writing therein :5, 17 = (.73, 3/) or = (.::+d.r, 3/+(l3/). The equation may be written l 77 — " I): ‘E (II I .r: ; and this form is applicable whether 3/ is given directly as a function of .’I.', or in whatever way 3/ is in effect given as a function of 2:: if as before .2‘, 3/ are given each of them as (ly . dy _ dz: . 3 (E 1S =d—9 ‘T (-13, which is the result obtained from the original form on writing (la: cly L72 ,7: being now the differential coefficient of 3/ in regard to a function of 6, then the value of therein for clr, (l3/ respectively. GEOMETRY [i~L.-mi: .-‘.'AI.YTICAL. So again, when the curve is given by an equation 21 = 0 . (ll _ . between the coordinates (.r, y), then is obtained from .7} . du die dy the equation (,3 + (-11; (E — using the original form, to eliminate zl.r, ¢l_:/ by the formula i ‘ii: tlx + But here it is more elegant, d . . (T: ¢l_:/ ; we thus obtain the equation of the tangent in the form d I . d—;5(£-96) +:,—_:,‘«n—2/;=')- For example, in the case of the ellipse + =1, the . . as q . . equation 1S ?(£ -1‘) + (17 -3/)=0_: or reducing by means of the equation of the curve the equation of the tangent is E 11;/_ a‘-‘+12’-’_' ' The normal is a line through the point at right angles to the tangent ; the equation therefore is (£—x)dx+ (11-2l)‘7!l= 0. where ¢l.r, (lg are to be replaced by their proportional values as before. 22. The circle of curvature is the circle through the point and two consecutive points of the curve. Taking the equation to be (5 ’ a)2+ (77 _ B)2=72z the values of a, B are given by = (73/(d.r'-’+d3/‘-’) (l.2:d‘-’_z/—(l3/(l'3.z: ’ y — and we then have — (7.r(rl.v:'-’ + (7)/‘-') ¢l..v.‘d'-’3/ —- cl}/ll‘-'.c ’ 0 _, _, (¢l.L"*’ + 0’_1/'-’)-3 7‘: = ‘ “l” + (-71 ' 3)" = ((1.1: (l"_1/ 1 (Z3/_ cl’-'31-}? In the case where 3/ is given directly as a function of .r, (l); ¢l‘-’1/ . ‘ ‘ this (la; ' 9 _ (l.::3 ’ then, writing for shortness 12 = is .. (1+212)'"’ 7" = 72; 1 (1+2v3)‘5‘ ” 9 positive or negative according as the curve is concave or coiivex to the axis of It may be added that the centre of curvature is the intersection of the normal by the consecutive norinal. The locus of the centre of curvature is the evolute. If from the expressions of a, /3 regarded as functions of .7: we eliminate .7", we have thus an equation between (a, ,8), which is the equation of the evolute. as the equation is usually written, , the radius of curvature, considered to be Polar C'u0rclinatcs. ‘.23. The position of a point may be determined by nieans of its distance from a fixed point and the inclina- tion of this distance to a fixed line through the fixed point. Say we have 1' the distance from the origin, and 6 the inclination of 'r to the axis of cc; 1- and 0 are then the polar coordinates of the point, 7' the radius vector, and 9 the inclination. These are immediately connected with the Cartesian coordinates or, 3/ by the formulae .2:= 1' cos 0, 3/=7‘ sin 0 ; and the transition from either set of coordi- nates to the other can thus be made without dilliculty. But the use of polar coordinates is very convenient, as well in reference to certain classes of questions relating to curves of any l;ind—for instance, in the dynamics of central forces—as in relation to curves having in regard to the origin the symmetry of the regular polygon (curves such as that represented by the equation r= cos 722(9), and
also in regard to the class of curves called spii-al.~'., where