NOTE ON THE NUMERICAL TRANSCENDENTS S AND s = - 1. n n n BY PROFESSOR W. WOOLSEY JOHNSON.

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1 NOTE ON THE NUMBERS S AND S =S l. 477 NOTE ON THE NUMERICAL TRANSCENDENTS S AND s = - 1. BY PROFESSOR W. WOOLSEY JOHNSON. 1. The umbers defied by the series 8 =l ' O 3 W 4 W } where is a positive iteger greater tha uity, are of frequet occurrece i aalysis. Euler, i the Istitutioes Calculi Differetialis, 1775, gave a table of their values up to 8 U to sixtee places of decimals (page 456). He had coected the values of the eve-umbered oes with Beroulli's umbers ad the eve powers of TT by the formula 2 ~~ (2)! ' but failed to obtai a fiite expressio for the odd-umbered oes. He gave also the value of the costat 7, ow kow as Euler's costat, defied as the limitig value 2. The costats 7 ad S occur i the expressios for the fuctio log T(l + x) ; ad Legedre, for the purpose of costructig his table of log T(a), examied Euler's table of values for 8 ad, fidig " quelques erreurs assez graves/' recostructed the table, carryig it to 8 S5, remarkig that for higher values of oe has oly to divide the excess over uity successively by 2. This is of course because the powers of 1/3, 1/4, etc., have disappeared from the last retaied decimal place, which was as i Euler's table the sixteeth. The values of 8 rapidly approach that of the first term, which is uity. It follows that, a algebraic series beig give i which the 8Js occur, if the series resultig from replacig 8 by uity has bee already summed, we ca by subtractio

2 478 NOTE ON THE NUMBEKS S AND S = S '\. [July, obtai a much more rapidly coverget series. replaced his series Thus Legedre TTX log r(l + x) = J log^^- yx - i8^- IS^ by. _ ^.,. TTX.,. 1 + X log r ( i + x) = j log^^ - j io g j ^ + (]- 7 )«; -K'S3-iK-K^-iK--- ; 3. I propose i this ote to put 8 = 1 + s, so that S = 4-1 f A table of values of 8 is of course virtually a table of values f * w ' Whe values of x for which the T-fuctio is kow are substituted i the algebraic series, umerical series ivolvig S ad 7 are foud ; thus Legedre * derives from the above equatio by puttig x = 1 ad x = J, (1) 1-7 = 1 log 2 + i«s + ^ + ik +. -, (2) i _ 7 _ l o g + ^ + ^ + ^ Legedre computed 7 from each of these series by meas of his table of S 9 ad cites the agreemet with the kow value of 7 as a test of the accuracy of the table. Dr. Glaisher i a paper ' ' O the history of Euler's costat," Messeger of Mathematics, 1871, cites a umber of these relatios from the Memoirs of Euler, oe of which is, i the preset otatio, (3) 7 = & + f», + *«* + - This formula was give i 1769, ad i a memoir of 1781 occurs the formula (4) 1-7 = K + i«8 + K + * Traité des foctios elliptiques et des itégrales eulériees, vol. 2, p The table of values of 8 a is o p. 432.

3 1906.] NOTE ON THE NUMBERS S AND s =S l It was a compariso of these equatios which first led me to otice the very simple relatio, idepedet of 7, (5) 1 = s 2 + s s + s This relatio must doubtless have bee frequetly oticed by those who have had occasio to deal with these umbers,* yet it seems clear that it was ot kow to Legedre writig i 1826, for it forms a much better verificatio of his table (ivolvig, as it does, every figure of it) tha does the computatio of 7 metioed above. 5. It is the mai object of this ote to poit out that ot oly this result but the other results metioed above ivolvig 7 ad aperia logarithms may be derived by direct summatio of series i s 's. Thus, if we write out the terms of each s i a colum, we have \-(*) +(*)*+(*)* + The rows form geometric progressios, hece we have 1 I like maer we ca show that whece it follows that S 2 ^~ S B + S 4 "~ ' ' * ^ Ï > (6) * 2 + s, + s & + - f, * I have, sice writig the above, foud it give as a problem i the secod editio of Boole's Fiite Differeces i the form : " Shew that the sum of all the egative powers of all whole umbers (uity beig i both cases excluded) is uity ; if odd powers are excluded, it is }."

4 480 NOTE ON THE NUMBERS S AND S = S 1. [July, ad (7) s s + s 5 + s r + = J. Legedre's table is completely verified by these equatios, the sums beig i fact oly 2 or 3 uits out i the 16th place. Agai, to sum s 2 -f Js 3 + -, we have 2 M + my + *(*) + + m 2 + mr + Here each row is of the form 1 \3? + \x* + \X* + = log l X where x has the values J, J, J, etc., i the successive rows. Hece Ê S ~ = [log 2 + log + log + + log ^ j which is equatio (4) above. 6. Treatig the sum s 2 + ^s à + ^s 6 +. i the same maer, the rows are of the form ^ + K + K + = lo gï"z^2' 7, x havig the successive values J, J, J, etc. Thus (8) -log$ + logf + logh yl 1 = log[f I lî!f-]=log2.

5 NOTE ON THE NUMBERS S AND $ = # As a further example, Legedre's equatio, (2) of this ote, o a may be verified. take the form, if y = Jœ, Thus i summig y^ + ^~ i +, the rows x^ oc?* 'xj P 'î/^ v^ ~\ 1-4-?/ ^+5^+7TF= 2 [ï J =l0g r^- 22/ where y takes the successive values ^, ^, -, etc. Hece *-*(2 + 1Y2 2 " LF ^ T 2»-i_U=«= [log(2+ 1)- log L2" + 3" + + ' * ' + ^:Jw =oo ad, sice log (2 + 1) at the limit becomes log 2 + log, th«result is which is equatio (2). 7. I have bee tempted to make a idepedet calculatio of the values of s, usig 12 decimal places. The values of s whe > 9, whe carried thus far, are readily obtaied by direct summatio of all their terms. But the extreme slowess of the covergece of the terms whe is small reders it practically ecessary to employ the same method as that by which 7 has bee computed, amely the Euler-Maclauri formula for the summatio of a fiite umber of terms, viz. : C, B.du B^d?u BA b u J * * x 2 dx 4! dx B ^ 6! du 5 I the applicatio of this formula whe u x is a egative power of x, the costat O is the sum to ifiity, ad is obtaied by direct summatio of Su x to a moderate value of x, ad calculatio of the ifiite series i the secod member. It was thus that Euler ad Legedre calculated the value of y as well as the values of S J ad thus that Adams, after greatly exted-

6 482 WRONSKIANS AND RELATED MATRICES. [July, ig the rage of kow Beroullia umbers, calculated 7 to 263 place of decimals. Iasmuch as Legedre's table has ot ofte bee reprited, it may be of iterest to give the results of my computatio to the eleveth place of decimals. They are as follows : Values of s = ^ + ^ s [The values for > 25 are obtaied each by dividig its predecessor by 2.] ANNAPOLIS, May, ON CERTAIN PROPERTIES OF WRONSKIANS AND RELATED MATRICES. BY PROFESSOR D. R. CURTISS. (Read before the Chicago Sectio of the America Mathematical Society, April 14, 1906.) I this ote I shall preset theorems of a very geeral character o the vaishig of Wroskias ad related matrices. Proofs, however, will be reserved for subsequet publicatio i more exteded form. Let u v u 2, -, u be fuctios, real or complex, of the real variable x, havig fiite derivatives of the first h orders