On equating coefficients of I, I, 13, I2 in (2) and (12) we find in succession A 23 22 =230.3.5.7.13.12 75585 62467 32499. As a verification these numerical values are found to satisfy the second and third of the equations (11). The numerical equations obtained by comparing coefficients of Jo, Jo, J', Ï, Jo in (10) and (2), and of 15, 1o in (12) and (2), are also found to verify accurately as congruences (mod 7), (mod 11), (mod 13) and (mod 17). The final form of F(J, I) is Jo + 1 o − J3 13 +23.3.5.31. (J3 1* + J‘13 ) - 2.3.5.131.193. (J3l3 + J3 13) +25.53.13.1 95053. (J3I2 + J'l3) -2.3.5.16445 56073: (J3I + J13) +23.5.83299 96823.J*1* +2.3.5.4494 95534 10557. (J^13 + J31*) - 2.5.11.17.23593 95537 50221 83703.J3]3 + 2.3.5.116.31.1193. (J + I) +290.31.53.11o. In these coefficients the six-figure factors and also the smaller factors are primes. I have made no systematic attempt to factorise the twelve remaining numbers between 10° and 10, but I have found that they are divisible by no prime less than 40. 6. The genus of the equation F (J, 1) = 0 is zero. In fact F. Klein has shown* that and TT125, 7 being an auxiliary modular function for quintic transformation. The relation F (J, 1) = 0 could also have been obtained by elimination of 7, 7' from Klein's equations. CALCULATION OF THE FIRST THIRTY-TWO EULERIAN NUMBERS FROM CENTRAL DIFFERENCES OF ZERO. § 1. A By S. A. JOFFE. SYSTEMATIC calculation of the first twentyseven Eulerian numbers was recently accomplished by Dr. Glaisher in an article "On Eulerian Numbers " in which he described the method of computation and established congruence-formulæ for testing the correctness of the results. The method consisted in the application of certain recurring formula which Dr. Glaisher had previously developed, and the process was shown to be especially advantageous when "curtate" formulæ were employed. The values obtained by Dr. Glaisher have now been independently verified, and to his list five more Eulerian numbers have been added, by a different method based upon a formula which expresses an Eulerian number as a function of "central differences of zero." The following contains. a description of the method, with its application to the calcu "Ueber die Transformation der elliptischen Functionen und die Auflösung der Gleichungen fünften Grades," Math. Ann. xiv, S. 111–172. † Quarterly Journal, vol. xlv., pp. 1-51. lation of the first thirty-two Eulerian numbers from auxiliary tables of "central differences of zero." § 2. As is well known, the Eulerian number E is the coefficient of x/(2n)! in the development of secx as a power-series of x, that is, E is defined by the equation. " .(1). as an The connection between E and the central differences of zero was shown by Mr. S. T. Shovelton, in a paper "Generalization of the Euler-Maclaurin sum-formula incidental consequence of a formula for secx obtained from the properties of the Bernoullian function. With the symbol 8' used to denote the finite-difference operator ▲'E-1, Mr. Shovelton's formula (38, last line) is as follows: E=(−1)" [1 − 183 + 48′ — 18° +...] 02”...................(2). The same formula was given by Dr. W. F. Sheppard† in a somewhat different form, and was derived by him from a study of the function n! F ̧ (x) = x" — E1 2! (» − 2)! n! * 4! (n−4)! by taking the last term of the expansion for n = 2r. ...(3), §3. In view of the fact that this formula is the foundation of the method employed in the present paper, it seems appropriate to deduce it without reference to the text of preceding writers. This may conveniently be done by applying Herschel's Theorem, which is to the effect that if f(e') is expressed as a power-series in t, say, then the coefficient of t/n!, A, equals ƒ(1+▲) O". Since the formula expressing secx as a function of e* contains imaginary quantities, we may, in order to avoid imaginaries, replace the circular function by the hyperbolic, and equation (1) will then become (1+A) +(1+A) ̄12 = (1 + ▲) + (1 − ▲′) ( 1 + ▲) ̃' + A2 (1 + A)−1 therefore, where the successive terms 83m On are the "central differences of zero," all of even order. For the purpose of emphasizing the property just established in formula (5) that every Eulerian number E, may be obtained by alternately adding and subtracting the central differences of O", affected by the factor we shall introduce 1 2m 1 the symbol e to denote the expression SO, and with this notation we shall have 2m §4. The central differences of zero of even order are connected by the recurring formula 82m (3n = m2 8TM O22 + 2m (2m − 1) 8am-3 ()11-3, and if we divide this by 2", we obtain ΟΙ 1 2m 1 2" 8am (2n = m2 (28TM O-1) + m (2m − 1) (1 em, n = m3 em, n-1+m (2m — 1) em-1, n-1 • • The arithmetical process of computing the successive e's from this recurring formula will be shortened considerably if we write the latter in this form: em, n = m[(m − 1) em-1, n-1 + mem-1, n-1 + mem, n-1]...... (8). ท.ท By referring to the following table, where all the details necessary for the computation of e, are given for the first seven values of n, it will be seen, for instance, that the results for n=7 are obtained from those for n=6 as follows:-The numbers in the last column for n=6 are multiplied respectively by 1 and 2, 2 and 3, 3 and 4, ...; say, 1,580,040 is multiplied by 4 and 5, and the products, 6,320,160 and 7,900,200, with the other similar products constitute column 1 for n=7. Each number in column 2 is the sum of the corresponding number in column 1 and the next two numbers above it; e.g. 45,405,360 is the sum of 6,320,160 +7,900,200 + 31,185,000. Finally, the numbers in the third column are the products of the corresponding numbers in the second column by 1, 2, 3, ... respectively. For instance, 227,026,800 = 5 x 45,405,360. 5 567 000 1 247 400 6 237 000 31 185 000 45 405 360 227026 800 680 400 6 7 37 422 000 1 247 400 7 484 400 44 906 400 113513 400 681 080 400 52 390 800 97 297 200 681080 400 |