and if in Ex. (27) of Chap. II. Sec. 1, we make r = 2n − 1, These coefficients B1, B3...B2n-1, are of great use in the expansion of series, and bear the name of Bernoulli's numbers, having been first noticed by James Bernoulli in his posthumous work the Ars Conjectandi, p. 97; but the complete investigation of the law of their formation is due to Euler, Calc. Diff. Part II, Cap. V. (10) To expand tan 0 by means of the numbers of Bernoulli The coefficient of 02-1 in the expansion of this function will be the same as that of an in the development of € +1 multiplied by 22 (-)". By what has preceded it appears, therefore, to be equal to (11) To expand cot by means of Bernoulli's numbers. Now the coefficient of 0-1 in this expression is the same CHAPTER VI. EVALUATION OF FUNCTIONS WHICH FOR CERTAIN VALUES OF THE VARIABLE BECOME INDETERMINATE. IF u be a function of a of the form P and if for Q the value a = a, P and Q both vanish; u, taking the form 0 is indeterminate and its true value will be found by differentiating the numerator and denominator separately and taking the quotient of these differentials: that is, using Lagrange's notation, the real value of u will be But if the same value (a = a) which makes P and Q vanish also make P′ = 0, and Q' =0, we must differentiate again, and so on in succession, as long as the numerator and the denominator both vanish when a is put equal to a. Therefore we may say generally that the true value of u when x = a is P) and Q" being the first differential coefficients of P and Q which do not vanish simultaneously when x is put equal to a. This theory of the evaluation of indeterminate functions was first given by John Bernoulli, Acta Eruditorum, 1704, p. 375. which when a = 1 is equal to n, as we have just found. This was one of the first functions the value of which was determined in this manner. One of the most important applications of this process is to find the sums of series for particular values of the variable. The first example was an instance of this, and we shall here add others taken like that from Euler's Calc. Diff. P. 746. and we are required to find its value when x = 1, or when the series becomes that of the first order of figurate numbers. By two differentiations we find 2 x + x2 − (n + 1)2x2 + 1 + (2 n2 + 2 n − 1 ) x" + 2 — n2 x” + 3 and we are required to find the real value of u when x = 1, which in this case is the sum of the squares of the natural numbers. Differentiating three times we have |