544 APTENDIX. B. 



lues of y; u, u , u , u , and u , corresponding to tbe 



12 3 4 



values of Xf 0, j, 2, 3, and 4; or a=u; 



h + c-\-d+e—u — az:zu — w; 

 1 1 



2b + 4.c + Sd-\-l6e=u -u; 



2 



Sb + 9c-\-27d-{-Sle=u —u; and 



3 



4b + 16c-\-64:d+256ezzu —u: we may 



4 



here get the second series of equations most easily by multi- 

 plying the first by the coefficients of e, whence 



16& + 16c + 16c? -I- 16e= \Qu — 16m ; consequently 



1 



14&4-12cH-8c?=:16w —15m—?/ ; 



1 2 



and in the same manner 



78& + 72c + 54c?=i81m —80m — m ; and 



252& + 240c + 192c/=:256m — 255m— m . 



1 4 



Here the coefficients of c are obviously the most manage- 

 able, and they afford us 66 — 66?— 15m — 10m— 6m H-m , 



1 2 3 



and 286— 32c?=: 64m —45m— 20m +m ; then taking f^ of 



1 2 4 



the latter from the former, we have f6=3M — ffM — ^m H-m 



1 2 3 



— -^u ; and 6=4m — f|M— 3m -hfw — -Jm ; which agrees 



4 1 " 2 3 4 



with the result obtained, from the inversion of Taylor's the- 



dM' 

 orem, for — - A : and this method, though less elegant, has 

 da; 



the advantage of being more readily applicable to the case 



of ordinates not equidistant. 



HowLKTi aud Brimmkr, 

 Printers, lo, Frith-street, Solio. 



