METHODS OF INTERPOLATION. 333 



The fourth difference of these terms is — 



and consequently — 



-\.S{U2+ns)-{Ui+i(s)\ 



If we suppose that the series Ui, «3, ^(3, &c., is of an order not higher than 

 the third, the adjusted series u'3, u\, n'^, &c., will be of the same order, 

 so that its fourth dititerences will be zero, and both members of the above 

 equation will be equal to zero. But if each of the terms Wi, 112, &c., is 

 liable to an accidental deviation or error, whose probable amount is 

 denoted by s, then the probable value of J4, taken ^vithout regard to 

 Bign, will be — 



(^4)=^7(«^--34)'+2[(47c-32f+(A:-22)2+82+l] 

 which reduces to — 



Eegardiug (J4) as a fnnctiou of the variable k, we have the equation — 



dk 



from which to find that value of A; which makes (J^) a minimum. This 

 is A-=-UJ- ; and substituting it in (G8), we obtain — 



«3=Tk[lll«3+56(%+W4)-14(wi+W5)] . . (69) 



which is thfe adjustment formula sought. 



To find a similar one including seven terms, we may take the most 

 general form as used in obtaining (52), or, what amounts to the same 

 thing, by proceeding as in the demonstration of formula (20), we can get — 



^*=/7qri(r^i:35[(''+4''^-i^)"''+('i'^*-i^)("3+^<5) 



Since k' affects only the weight of the middle term, we may, for the sake 

 of brevity, denote that weight by k' alone, and so write — 



The expression for the fourth difference of the adjusted series then is — 



+ (^•'-22 /v-+100)(«4+W8)-(45-8 k){i(,+na) 

 + (10-Z;)(«2+Mio)-(Wi+Wn)J 



