5. The series discussed in the preceding article admit also of the method of treatment developed in the treatise on Differential Equations (p. 435), and it is very interesting to compare the modes in which the same conditions of finite algebraic expression present themselves in solutions so totally distinct in form. The last example will serve as an illus tration. If we make x = =e, the series (a) becomes παθ (D + a) ̃1 $ (0) = € ̄«° D ̄1. €ao $ (0), and then restoring x, we have for the sum of the series the expression Now let x=4. Then substituting 4 for x in the terms without the sign of integration and in the upper limit, we have The integration can now be effected, and we have as the result But what is the ground of connexion between the two methods? It consists in the law of reciprocity established by the theorem of Chap. II. Art. 11, viz. (t) and (t) being functions developable by Maclaurin's theorem, and t being made equal to 0 after the implied operations are performed. To establish this let the proposed series be $ (a) + $(a + 1) + $ (a + 2) ... + $ (n). Its corrected value found by the method of this chapter is Σφ (η + 1) - Σφ (α), the summation having reference to n in the first and to a in the second term. Its value found by the other method is where, after the implied operation, e*= 1 and therefore t=0. d = (can — 1) ̄1μ (t+n+1)—(eda—1) ̄1$(t+a), since t is connected with n in the one term and with a in the other by the sign of addition. And now, making t=0, the expression reduces to Σφ (+ 1) - Σφ (α), which agrees with the previous expression. d Conditions of extension of direct to inverse forms. d d dx 6. From the symbolical expression of Σ in the forms (ex — 1)~1, and more generally of Σ" in the form (e2 – 1) ̃”, How certain theorems which may be regarded as extensions of some of the results of Chap. II. To comprehend the true nature of these extensions the peculiar interrogative character of the expression (1)", must be borne in mind. Any legitimate transformation of this expression by the development of the symbolical factor must be considered, in so far as it consists of direct forms, to be an answer to the question which that expression proposes; in so far as it consists of inverse forms to be a replacing of that question by others. But the answers will not be of necessity sufficiently general, and the substituted questions if answered in a perfectly unrestricted manner may lead to results which are too general. In the one case we must introduce arbitrary constants, in the other case we must determine the connecting relations among arbitrary constants; in both cases falling back upon our prior knowledge of what the character of the true solution must be. Two examples will suffice for illustration. Ex. 1. Let us endeavour to deduce symbolically the expression for Zu,, given in (3), Art. 1. Now this is only a particular form of Eu, corresponding to a∞ in (3). To deduce the general form we must add an arbitrary constant, and if to that constant we assign the value - (Ua_1 + Ua_2 ••• +&c.), we obtain the result in question. Ex. 2. Let it be required to develope Σuv, in a series proceeding according to Ev, v2, &c. We have Σuxvx = (DD' — 1)−1 UxVx, D referring to x only as entering into u,, D' to x only as entering into va =ux_1Σvx − Aux_2Σ2vx + A3ux_3Σ3⁄4vx — &c., the theorem sought. n-1 In applying this theorem, we are not permitted to introduce unconnected arbitrary constants into its successive terms. If we perform on both sides the operation A, we shall find that the equation will be identically satisfied provided Au, in any term is equal to Σu in the preceding term, and this imposes the condition that the constants in "u, be retained without change in "u. And as, if this be done, the equation will be satisfied, it follows that however many those constants may be, they will effectively be reduced to one. Hence then we may infer that if we express the theorem in the form Σuxvx = C + ux_1Σvx − Aux-2 Σ2vx + A2ux-2 Σ3v2 · (1), we shall be permitted to neglect the constants of integration, provided that we always deduce Σ"v, by direct integration of the value of "v, in the preceding term. If u be rational and integral, the series will be finite, and the constant C will be the one which is due to the last integration effected. 1 + a cos 0 + a2 cos 20...+a* ̄1 cos (x − 1) 0. 6. The successive orders of figurate numbers are defined by this, viz. that the th term of any order is equal to the sum of the first x terms of the order next preceding, while the terms of the first order are each equal to unity. Hence, shew that the ath term of the nth order is x(x+1)(x+2) ... (x + n − 2) 7. It is always possible to assign such values of s, real or imaginary, (being the roots of an equation of the nth degree) that the function |