STORY. — A NEW GENERAL THEORY OP ERRORS. 175 



Reversing the order of integration, observing that ^ — ^' must take 

 all values from ^ to | and that, by (19), i/' (|) = and i/^ (|) = 0, we 

 have 



f ¥+(' 



i' l+r 



f 1 



= JV (O • di' J\^ + er -^iO' di 

 i' I 



m m 



2 



where I . J denotes the coefficient of x* in the development of (1 + a:)"* 

 by the binomial theorem, that is 



(23) 



{■J {m-k)\kl 

 Therefore, 



TO 



In particular, for the smaller values of m, 



Spo = 0, 8pi = (because po = 1 find j»i = 0), 



8p2 = po', ^Ps = p/, 8p^ = Qp..pJ + ;>/, 



gps = lOpsp/ + \Op2p3' -\- p&, 



Spe = l^PipJ + ^Opspa' + lop.pi' + p^', etc., 



and, therefore, 



/>./ = S;)2 , p^ = 8ps , Pi' = 8p^ — Gp2- 8p. , 



P^J = S/>5 — 10 7)3 • 8po — 10p2 ■ ^ps, 



Ps = ¥c, — 15i^4 • Sy>., - 20 p^ ' S;?3 — 15;^., (Bp^ — GjOj " ^^"2) 



= Spe — 15j02 • Spi — 202>3 • ^JOg — 15 (i^4 — 6^2^) ^^'2 , StC. 



