1873.] Mr. W. Shanks on Discrepancies in the Value O/TT. 45 



III. " On certain Discrepancies in the published numerical value 

 of TT." By WILLIAM SHANKS, Houghton-le-Spring, Durham. 

 Communicated by Prof. G. G. STOKES, R.S. Received October 

 13, 1873. 



The author's attention has been recently drawn, by Mr. John Morgan, 

 of Bishopbriggs, Glasgow, to two cases of discrepancy hi the author's 

 published value of TT, and to the misprint of a figure, in the value of 

 tan" J -i- (one of the arcs employed in determining TT), as given in the 

 author's paper of April 1873. These two discrepancies in the value of 

 TT, which will be described presently, did not appear in the paper pre- 

 sented by the late William Eutherford in 1853, wherein the author's 

 extension to 530 decimals was correctly given. 



The source of the two cases of error was easily discovered, on re- 

 ferring to the author's manuscript of 1852 ; for it was there found that 

 the 461st and 462nd decimals in the value of tan" 1 ^ were, in the process 

 of careful revision previous to publication, altered from 88 to 96 ; and 

 this alteration required the addition of 128 in the corresponding place of 

 the value of TT. This addition was rightly made in the value sent to 

 Mr. Eutherford, and given, as above stated, in his paper. It should 

 seem that the author unfortunately did not take a copy (for his own use) 

 of what he forwarded to Mr. Eutherford. At all events, from some 

 strange cause or accident (perhaps from being overworked) the addition 

 of 128 was very soon afterwards made at the 513th, 514th, and 515th, 

 instead of at the 460th, 461st, and 462nd decimal places. Hence arose 

 the double error, which remained, strange to say, undetected for upwards 

 of twenty years ! This mistake, however, from its nature, in no wise 

 affects the accuracy of the other figures. 



It is sufficient merely to state that the 75th decimal, before alluded 

 to, in the value of tan~ l 1 should be 8 and not 7 ; but it may be well to 

 give the entire value of TT anew : 



Value of 7r = 3- 



14159 26535 89793 23846 26433 83279 50288 41971 69399 3751 58209 74944 

 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 06647 

 09384 46095 50582 23172 53594 08128 48111 74502 84102 70193 85211 05559 

 64462 29489 54930 38196 44288 10975 66593 34461 28475 64823 37867 83165 

 27120 19091 45648 56692 346.03 48610 45432 66482 13393 60726 02491 41273 

 72458 70066 06315 58817 48815 30920 96282 92540 91715 36436 78925 90360 

 QH33 535 48820 46652 13841 46951 94151 16094 33057 27036 57595 91953 

 09218 61173 81932 61179 3 I0 5 I '8548 07446 23799 62749 56735 18857 52724 

 89122 79381 83011 94912 98336 73362 44065 66430 86021^39501 60924 48077 ' 

 23094 36285 53096 62027 55693 97986 95022 24749 96206 07497 03041 23668 

 86199 5 IIO 89202 38377 02131 41694 11902 98858 25446 81639 79990 46597 



00081 70029 63123 77387 34208 41307 91451 18398 05709 85 &c. 



j** 



The statement in the author's paper of April 1873, touching Richter's 



