Tables connected with the Pellian Equation from the point where the work was left by Degen in 1817.-Report of the Committee, consisting of Professor A. CAYLEY, Dr. A. R. FORSYTH, Professor A. LODGE, and Professor J. J. SYLVESTER. (Drawn up by Professor CAYLEY.) WE have on the Pellian Equation Degen's tables, the title of which is 'Canon Pellianus sive Tabula simplicissimam æquationis celebratissimæ y=a+1 solutionem pro singulis numeri dati valoribus ab 1 usque ad 1000 in numeris rationalibus iisdemque integris exhibens. Autore Carolo Ferdinando Degen. Hafniæ, apud Gerhardum Bonnierum, MDCCCXVII. 8°. Introductio, pp. v-xxiv. Tabula I. Solutionem æquationis y-ax2-1=0 exhibens, pp. 3-106. Tabula II. Solutionem æquationis y-ax2+1=0, quotiescunque valor ipsius a talem admiserit, exhibens, pp. 109-112.' The mode of calculation is explained in the Introduction, and illustrated by the examples of the numbers 209, 173. As to the first of these the entry in Table I. is where the first line gives the expression of 209 as a continued fraction, viz., we have the denominators being 2, 5, 3, (2), 3, 5, 2, then 28, which is the double of the integer part 14, and then again 2, 5, 3, (2), 3, 5, 2, and so on, the parentheses of the (2) being used to indicate that this is the middle term of the period. The second row gives auxiliary numbers occurring in the calculation of the first row and having a meaning, as will presently appear. Observe that the 11 which comes under the (2) should also be printed in parentheses (11), but this is not done. The process for the calculation of the x, y is as follows: viz., writing down as a first column the numbers of the first row, and beginning the second column with 1, 14 (14 the number at the head of the first column), and the third column with 0, 1, we calculate the numbers of the second column, 29=2.14+1, 159=5.29 +14, 506=3.159+29, &c., and the numbers of the third column in like manner, 2=2.1+0, 11=5.2+1, 35=3.11+2, &c.; and then writing down as a fourth column the numbers of the second row with the signs +, alternately, we have a series of equations y-ax=±A, viz., the last of them being (46551)2-209(3220)2= + 1 this last corresponding as above to the value + 1, and the numbers 46551 and 3220 being accordingly the y and a given in the fourth and third rows of the table. As to the second of the foregoing numbers, 173, the only difference is that the period has a double middle term, viz., the entry in the Table I. is The first row gives the expression of √173, viz., that is the denominators being 6, 1, 1, 6, then 26 (the double of the integer part 13), and then again 6, 1, 1, 6, and so on. In the second row I remark that Degen prints the parentheses (13, 13) for the double middle term. The process for the calculation of the x, y is similar to that in the former case, viz., we have where the second and third columns begin 1, 13 and 0, 1 respectively, and the remaining terms are calculated 79 6.13+1, 92=1.79+13, &c., and 6=6.1+0,7=1.6+1, &c.; and then writing down as a fourth column the terms of the second row with the signs +, alternately, we have the term for the last equation being always in a case such as the present one, not +1, but —1. The final numbers 1118, 85 are consequently entered not in Table I., but in Table II., viz., the entry in this table is and thence we calculate the numbers y, z of Table I., viz., these are Generally Table II. gives for each value of a, comprised therein, values of x, y, such that y=a2-1, and then writing y1=2y2+1, x11 = 2xy, we have y12 = (2ax2 — 1)2 = 4a2x1 — 4ax2 + 1 = a. 4x2(ax2 − 1) + 1 = ax12 + 1 so that 1, y1 are for the same value of a the values of x, y in Table I. It is to be remarked that the heading of Table II. is not perfectly accurate, for it purports to give for every value of a, for which a solution exists, a solution of the equation y2=ax2-1. What it really gives is the solution for each value of a for which the period has a double middle term. But if a=a2+1, then obviously we have a solution y=a, x=1, and for any such value of a the period has a single middle term, viz., the entry in Table I. is The foregoing instances of the calculation of x, y in the case of the numbers 209 and 173 suggest a table which may be regarded as an extended form of Degen's tables; viz., such a table, from a=2 to a=99, is as follows: Specimen of extended form of Table in regard to the Pellian Equation. |