h h^2 [f]([gamma]) + -- [f]'([gamma]) + --- [f]"([gamma]) + ... = 0; 1 1.2 and then, if h be so small that the terms after the second may be neglected, [f]([gamma]) + h[f]'([gamma]) = 0, that is, h = {-[f]([gamma])/[f]'([gamma])}, or the new approximate value is x = [gamma] - {[f]([gamma])/[f]'([gamma])}; and so on, as often as we please. It will be observed that so far nothing has been assumed as to the separation of the roots, or even as to the existence of a real root; [gamma] has been taken as the approximate value of a root, but no precise meaning has been attached to this expression. The question arises, What are the conditions to be satisfied by [gamma] in order that the process may by successive repetitions actually lead to a certain real root of the equation; or that, [gamma] being an approximate value of a certain real root, the new value [gamma] - {[f]([gamma])/[f]'([gamma])} may be a more approximate value. [Illustration: FIG. 1.] Referring to fig. 1, it is easy to see that if OC represent the assumed value [gamma], then, drawing the ordinate CP to meet the curve in P, and the tangent PC' to meet the axis in C', we shall have OC' as the new approximate value of the root. But observe that there is here a real root OX, and that the curve beyond X is convex to the axis; under these conditions the point C' is nearer to X than was C; and, starting with C' instead of C, and proceeding in like manner to draw a new ordinate and tangent, and so on as often as we please, we approximate continually, and that with great rapidity, to the true value OX. But if C had been taken on the other side of X, where the curve is concave to the axis, the new point C' might or might not be nearer to X than was the point C; and in this case the method, if it succeeds at all, does so by accident only, i.e. it may happen that C' or some subsequent point comes to be a point C, such that CO is a _proper_ approximate value of the root, and then the subsequent approximations proceed in the same manner as if this value had been assumed in the first instance, all the preceding work being wasted. It thus appears that for the proper application of the method we require _more_ than the mere separation of the roots. In order to be able to approximate to a certain root [alpha], =OX, we require to know that, between OX and some value ON, the curve is always convex to the axis (analytically, between the two values, [f](x) and [f]"(x) must have always the same sign). When this is so, the point C may be taken anywhere on the proper side of X, and within the portion XN of the axis; and the process is then the one already explained. The approximation is in general a very rapid one. If we know for the required root OX the two limits OM, ON such that from M to X the curve