ON MATHEMATICAL TABLES. 



53 



a Tl = 0-5 + 0-47455 39561 71379 

 a', a v = 0-5 =F 0-37076 55927 99697 

 a", a" = 0-5 + 0-20292 25756 88698 



a, 



8, 

 2, 

 5, 



84434 

 44638 

 52559 



= 0-5, 



A = A« =0-06474 24830 



A' = A v = 0-13985 26957 



A" = A" = 0-19091 50252 

 256 



A "' = 1225 = °- 20897 

 Coefficients, except A'", given by — 



95918 36734 



8 ; log = 8-81118 93529, 



4 ; log = 9-14567 08421, 



5 ; log = 9-28084 0J093, 



7 ; log = 9-32010 38766. 



1859 

 16800 '' 



1573 i 7947 

 29400 U + 39200 



1 . 



Approximate correction = -.75579359 h xlv . 



It is obvious tbat tbe foregoing formulae give at once tbe roots of the 

 equations P"(a;) = 0, viz., 



For n = 2, tbe roots are ± 0-57735 02691 89626 ; 



n = 3, „ „ 



» 



n = 4, 

 n = 5. 



n 



n 



= 6, 



= 7, 



and 



± 0-77459 66692 41483 ; 



± 0-33998 10435 84865, 



± 0-86113 63115 94049 ; 



and 



± 0-53846 93101 05683, 



±0-90617 98459 38664; 



± 0-23861 91860 83197, 



± 0-66120 93864 66264, 



± 0-93246 95142 03152 ; 



and 



± 0-40584 51513 77397, 



± 0-74153 11855 99394, 



-1- 0-94910 79123 42760. 



As tbe functions ~P n (x) contain only powers of 2 in tbe denominators 

 tbe decimal values terminate, and in tbe following tables tbe complete 

 values are given. Two independent calculations were made, one by Mr. 

 Barrett Davis, and tbe other under tbe supervision of the reporter, by 

 whom they were compared, corrected, and differenced. As the values 

 in the tables are complete, the second, third,... seventh differences in the 

 respective functions were absolutely constant, and thus afforded an exact 

 verification. The calculations were performed seven years ago, and are 

 referred to in tbe Report for 1873, p. 170 ; the tables were not published, 

 however, as about that time the issue of a separate volume containing 

 these and some other tables which it was proposed to calculate was con- 

 templated by tbe Committee. 



The plate (Plate I.) contains drawings of the curves y = P l (%), 

 y = V 2 (x),...y = P 7 (a;) from x = to x = 1, made from the tables. 

 The positions of the roots of the functions are readily identified with tbe 

 numerical values given above. 



