INFLECTION, or point of inflection, in the higher geometry, is the point where a curve begins to bend a contrary way. See FLEXURE.
There are various ways of finding the point of inflection ; but the following seems to be the most simple. From the nature of curvature it is evident, that while a curve is concave towards an axis, the fluxion of the ordinate decreases, or is in a decreasing ratio, with regard to the fluxion of the abscess; but, on the contra ry, that the said fluxion increases, or is in an increasing ratio to the fluxion of the absciss, where the curve is convex to wards the axis ; and hence it follows that those two fluxions are in a constant ratio at the point of inflection, where the curve is neither concave nor convex. That is, if x = the absciss, and y = the ordinate then x is toy in a constant ratio, or -or- is a constant quantity. But con stant quantities have no fluxion, or their fluxion is equal to nothing ; so that in this case the fluxion ofd or ofiis equal to nothing, And hence we have this general rule : viz. put the given tion of the curve into fluxions ; from x y or -; then take the fluxion of this ratio or fraction, and put it equal to 0 or no thing ; and from this last equation find also the value of the same or y: then put this latter value equal to the former, which will be an equation, from whence, and the first given equation of the curve, x and y will be determined, being the absciss or ordinate answering to the point of inflection in the curve. Or, putting the fluxion of - equal to 0, that _ = 0, or x: = 0, or ./.. y = Y, or 5 : y : : 5 : y, that is, the second flux ions have the same ratio as the first flux ions, which is a constant ratio ; and there fore if 2": be constant, or x = 0, then shall = 0 also ; which gives another rule, viz, take both the first and second flux ions of the given equation of the curve, in which make both x and y = 0, and the re sulting equations will determine the va lues of x and y, or absciss and ordinate, answering to the point of inflection.
To determine the point of inflection in curves, whose semi-ordinates C M, C m (Plate Miscel. VII. fig. 13 and 14.) are drawn from the fixed point C; suppose C M to be infinitely near C ni, and make mH=--.Mm; let T tn touch the curve in M. Now the angles C m T, C M m, are equal ; and so the angle C m H, while the semi-ordinates increase, does decrease, if the curve is concave towards the centre C, and increases, if the convexity turns towards it. Whence this angle, or, which is the same, its measure, will be a mini mum or maxmium, if the curve has a point of inflection or retrogression ; and so may be found, if the arch T H, or fluxion of it, be made equal to 0, or infinity. And in order to find the arch T H, draw an L, so that the angle T m L be equal to m C L ; then if C m = y, m r x, e, we shall have y : : : t: Again,draw the arch II 0 to the radius C H ; then the small right lines an r, 0 H, are parallel ; and so the triangles 0 L H, L r, are similar ; but because H I is also perpendicular to in L, the triangles L H I. m r, are also similar : whence t : x : : y ; at: ; that is, the quantities In T, m L, are equal. But H L is the fluxion of H r, which is the distance of C m = y; and II L is a negative quantity, because while the ordinate C M increases, their ence r 1:1 decreases; whence X x+ y y 0, which is a Feneral equation for finding the point of inflection, or re trogradation.