28. COR. 9. The expansion of (a±x)TM is a For (Art. 26.) when unity is divided by the coefficients of the different powers of x in the quotient can be reduced to the form the coefficients of the several powers of x in the quotient will be the same as before, except that m n must be written for m; wherefore, these coefficients may be reduced to the form 29. COR. 10. Thus, whether the index m of (a+x) be positive or negative, integral or fractional, it may be shewn, that the coefficients of the expansion of (a + x)" are formed according to the same law; and, that in no case of the expansion of a binomial, is the ratio of the latter of two consecutive coefficients, to the former, greater than any that can be assigned: And, therefore, the value of x may be taken such, that the difference between any one term, and the sum of all that follow it, shall be less than any given finite quantity. For, let p and q be any two consecutive coefficients; it has been demonstrated, in every case of the expansion of (a+x), that and, ifr be greater than m, it is less than 2. Therefore the ratio of q to p can never be so great as that no ratio can be assigned which is equally great; and, therefore, (Art. 6.) the value of x may be taken such, as that any one term of the expansion shall become the limit of all the terms which follow it. ON MAXIMA AND MINIMA. PART II. SECTION H. ON THE EQUATION WHICH SERVES TO DETERMINE THE VALUE OF ANY FUNCTION OF ONE OR MORE VARIABLE QUANTITIES, WHEN IT IS A MAXIMUM OR A MINIMUM. PREVIOUSLY to the estimation of continued quantity, it is necessary to make some hypothesis respecting the generation of variable magnitudes. BARROW enumerates eight different modes in which quantity may be supposed to be generated. Its increase and decrease by motion, which is the foundation of the doctrine of Fluxions, is readily conceived in a vague and general manner. But there is no inconsiderable difficulty in deducing, logically, from that primary notion, the rules of algebraic computation, without which mere theory is of little value. Motion implies velocity; velocity requires the consideration of time; and to any enquiry concerning the nature of time we are not yet enabled to return a much more satisfactory answer than that of AUGUSTIN, so often cited, "Si nemo quærat, scio; si quis interroget, nescio." All that seems necessary, in the first instance, in the place of the fluxional hypothesis, is to express, in as general a manner as is possible, the condition of a quantity being variable according to a certain law; i. e. of its admitting any variation whatever with respect to magnitude, so as to continue to be the same species of magnitude, as it was before that variation took place. If % denote a line variable at pleasure, and z' any addition or diminution of which it is capable, then + ' will denote the condition of that line: and if azm denote a surface, or a solid, variable at pleasure, but always retaining its species, expressed in terms of a variable line z, a (≈ ± x')” will be a general representative of its value. Of this kind is the notation adopted in the following section. There is little real difference, in the methods used by the ancient and modern authors, who have treated the investigation of Maxima and Minima algebraically. The reasoning of LAGRANGE has been principally followed, in the most important propositions belonging to this part of the subject. His rules of computation are the same with those of LEIBNITZ, and all the writers on Fluxions. It was the demonstration only of these rules, and not the rules themselves, which needed to be improved. 30. DEF. A quantity is said to be constant, when it is always of the same value throughout any calculation: A variable quantity is that which may have any value within certain limits: An arbitrary quantity is that of which the value may be supposed, in any calculation, either greater or less than any given finite quantity: In the language of Algebra, a function containing only one variable quantity, is called a Maximum, when the variable quantity can neither be increased nor diminished, without the value of the function being thereby made less than it was before: And it is called a Minimum, when the variable quantity can neither be increased nor diminished, without the value of the function being thereby made greater than it was before. 31. DEF. Variable quantities being denoted by the last letters of our alphabet, the same letters, with an accent placed above them, are used to express arbitrary quantities, by which those variable magnitudes are supposed to be increased or diminished. |