and the continued surd which is the inverse function may be similarly expressed. III. To expand the function a , a, the indices being continued a times. Denoting this quantity by u., we have 20-4- 1 = a . (z"* = q) (u.), and adapting the general formulae to this case, we have For the inverse function if we put a = }, and take logarithms in Before leaving this example, we may observe, that the most rapid way of finding A, is this, let log. 3 = b, i.e. a' = 3; and since I A. B-" > take its log. and add to b for a second; take the log. of the second approximation, and add to b for a third ; and when b > 1, we shall get a very converging series of values for A. A = ... A = b + log. A ; take b then as a first approximation, If we had applied Lagrange's theorem in this case to the equation A = a +", we should have A expressed in a divergent series; it is necessary therefore to limit the announcement, that this general series gives the least root, to the case of real roots, for when there are some imaginary, we see that it may express one of these instead of the least real root. IV. By a similar process we easily obtain " (mo - 1) (m = 1) : I.2. 3. 4.5 the value of B being the same in both, and found as before. CASE OF FAILURE OF THE GENERAL SOLUTION. When the equations op (A) = A, p'(A) = 1 are simultaneously true, the terms in the expansion of u, become infinite, as before remarked, we shall therefore give a solution in a different form, for this case. The equation p (A) = A must here have two equal roots, suppose each = c : transform the proposed equation by putting u, = r. -- c, the corresponding equation A = F(A) has then two roots each equal to zero, and consequently F(A) must be of the form Hence if v, vanish, v, v, v, &c. successively vanish, and therefore we put generally in which series the coefficients are unknown functions of a, but independent of vo. Beside the foregoing form of expressing r. 1, there are two others, viz. and since F is manifestly unity, if we compare the expressions 1 and 2, when the latter is arranged according to the powers of r, we obtain A. = 1, which is obvious by the law of the successive formation of the quantities r1, r., &c.; also putting a = 0 in the general value of v., which shews that B, - 0, C. - 0, &c.; this being premised, we have by comparing the expressions (1) and (3), the following identity; Before leaving this class of equations, we may remark a curious relation between the equations if then we determine u, so that v, = f'(u), v, will be readily found from u. : the equation p = 0 (putting A = A, = ......A., 1) determines A, T. = 0 will determine c, and B will remain an arbitrary constant. Now in linear equations the sum of the particular solutions gives the complete integral, but this is not generally the case in other in |