STORY. — A NEW GENERAL THEORY OF ERRORS. '195 



therefore, 



(95) i?t = for s = ± » ; 



therefore, e(iuafions (81) are all satisfied, by (89)-(91). Also, (92) and 



(93) become, for this case, 



-f-oo -|-oo 



fs.,„ Ro„, ■ ds = 2m fs,,„,, /?.,„_! • ds, 



00 —00 



+00 -Hjo 



Js,... ^=.« • * = ^^ /5.. A. • * ; 



00 00 



by alternate applications of these two formulae, we obtain 



+00 +c 



JS„„ /?„„ • ds = ^l^Js,„,_, i?.„,_2 • ds, 



— oo — oc 



+00 +00 



2 m + IJ 



— oc 00 



and again, by repetitions of these, 



+ » +00 



J S^^ i?2,„ • ds = . _ ,w„.) / -^0 ^0 • (^S> 



— » — » 



+00 4-00 



— 00 



Now, by (93), (63), and (68), 



+X +» +00 ' 



jSx R,-ds = jSo Ho ■ ds = fe"'^ • ds = a/2^ 



00 —^ X 



where tt represents the ratio of the circumference of a circle to its diame- 

 ter ; therefore, 



+^ 



(96) J^,„. i?,„ • ds = (^,^^, V2^ 



1. 



+00 



(97) / ^2,„+i i?2,«+l • (^ = ^.„, Z \\i'"> ^^ ^' 



m ? 



(»*Ti)" 



