to the form a y2+ b x y + c x2 + ƒ= 0, or, finally, to the form a y2 + c x2 + ƒ = 0. Although, however, we cannot thus reduce the parabolic equation, we are yet able to reduce it to a very simple form, in fact to a much more simple form than that of either of the above equations. This will be effected by a process similar to that already used for the general equation, only in a different order. We shall commence by transferring the axes through an angle 0, and thus destroy two terms in the equation, so that it will be reduced to the form a y2+ dy + ex + f = 0; we shall then transfer the axes parallel to themselves, and by that means destroy two other terms, so that the final equation will be of the form substituting these values in the general equation a y2 + bx y + c x2 + dy + e x + ƒ = 0, and arranging, we obtain the equation a (cos. 4) -b sin. cos. + c (sin. )2 y2+2 a sin. cos.xy+a (sin. 6)2x2 + d cos. y′+dsin. | x' +ƒ=0 +b (cos. 4)2 + b sin. cos. - e sin. @ +e cos. A - b (sin. 4)2 +c (cos. 4)2 - 2 c sin. cos. | Let the co-efficient of x' y' = 0 .. 2 (a–c) sin 0 cos. ℗ + b {(cos. 0)2 — (sin. 0)2} = 0, Hence, if the axes be transfered through an angle e such that tan. 2 0 = a b C the transformed equation will have no term containing the product of the variables; that is, it will be of the form a' y12 + c' x12 + d' y' + e' x' + f = 0. But, since this last equation belongs to a parabola, the relation among the co-efficients of the three first terms must be such that the general condition be 4 a c 0 holds good. In this case, since b' = 0, we must have 4 a'c' 0; hence either a' or c' must = 0; that is, the transformation which has enabled us to destroy the co-efficient of the term containing 'y' will of necessity destroy the co-efficient of either x'2 or y'2. And this will soon be observed upon examining the values of the coefficients of x and y'2. 93. Let the co-efficient b in the original equation be negative, that is, let b = 2 √ac. since sin. 2 must be positive, and b is itself negative; hence cos. = 1 + cos. 20 a 1+ a + c = a a + c Substituting these values of sin. O and cos. O in the general transformed equation, we have And it is manifest that if b had been positive all the way through this article, the reduced equation would have been de (a + c) x12 + - e√ a √a+c y' + c ·d √ a + e √ © x2 + f = 0. √atc + 94. In order to reduce the equation still lower, let us transfer the axes parallel to themselves by means of the formulas y' = y" + n and x' = x" + m (54.) then the equation a' y' + d'y' + e'x' + f = 0 becomes a' (y" + n)2 + d' (y" + n) + e' (x" + m) + f = 0, or a' y2 + (2 a' n + d') y" + e' x'' + a' n2 + d' n + e' m + f = 0. And since we have two independent quantities, m and n, we can make two hypotheses respecting them; let, therefore, their values be such that the co-efficient of y' and the constant term in the equation each = 0; that is, let 2 a' n + d' = 0, and a' n2 + d' n + e' m + f = 0; and it is manifest that if b had been positive, the equation ca12 + d'y' + e'x' +ƒ0 would have been reduced to the form where the values of m and n would be found from the equations 95. The following figures will exhibit the changes which have taken place in regard to the position of the locus corresponding to each analytical change in the form of the equation : In fig. 1, the curve is referred to rectangular axes A X and A Y, and the equation is a y2 + b x y + c x2 + dy + e x + f = 0. - In fig. 2, the axes are transferred into the position A X', A Y', the b angle X A X' or being determined by the equation tan. 2 0 = athe corresponding equation is, for b negative, a' y'2 + d' y' + e' x' + ƒ = 0. If b is positive, the curve would originally have been situated at right angles to its present position, and the reduced equation would be c' x'2 + d'y' + e'x' +ƒ= 0. In fig. 3, the position of the origin is changed from A to A', the coordinates, of A' being measured along A X' and A Y', and their values determined by the equations 96*. If the original axes are oblique, the transformation of the general equation must be effected by means of the formulas in (55). The values of a', b', and c' will be exactly the same as in (87). We may then let b' 0, and also find tan. 20 when the axes are rect *See Note, Art. 87. angular, whence, as in (87), we shall find that there is but one such system of axes. The same value of 0 which destroys the term in x'y' will, as in (93), also destroy the term in a22 or y22; hence the reduced equation will be for c sin. 2 w for c sin. 2 w b sin. w positive, a' y'2 + d' y' + e' x' 97. To find the values of a', c', d', and e'. +ƒ = 0. + ƒ = 0. The values of a' and c' are best deduced from those in art. (88), 0, we have for c sin. 2 w — b sin. w positive Since b 4 a c (d - sin w e cos. w) √ { a − b cos. w + c (cos. w)2} —e √c (sin. w)2 and the reduced equation is now of the form a' y'2 + d' y' + e' x' + ƒ = 0; For c sin. 2 wb sin. w negative, the corresponding values of a', c', M, d', and e' are hence d' = and e' = (d- — e cos. w) √c-e √ { a−b cos. w+c (cos. w)2 } √ {a - b cos. w + c} (d-e cos. w) ↓ {a−b cos. w+c (cos. w)2}+e ↓c (sin. w)2 and the reduced equation is now of the form c' x'2 + d' y' + e' x' + ƒ = 0. The transformation required to reduce the equations still lower is performed exactly as in (94); and, by making the angle between the original axes oblique, the figures in (95) will exhibit the changes in the position of the curve. 98. We shall conclude the discussion of this class of curves by the application of the results already obtained to a few examples. Ex. 1. y 6xy+9x2+10y+1=0; locus a parabola. tan. 29 Ex. 3. y + √x = √d. This equation may be put under the form y+x d = 2xy; or and the locus is a parabola because it satisfies the condition By tracing the curve as in (78) we shall find its position to be that of the last two quantities are to be measured along the new axes, therefore Ex. 4. yd+ex+fa. The locus is a parabola, since 62-4 ac or 0 4.0. f = 0. F |