278 METHODS OF INTERPOLATION". 



83 = 



Let S be a fourth area whose base is w, and let x' be the abscissa cor- 

 responding' to its middki ordinate 5 then — 





(1) 



Eliminating A, B, C, from the above four equations employing P, Q, B, 

 as auxiliary letters, ami dropping the accent from x'^ we have — 



Q = a,[^2^-j-V(;i='-«2')]+;r[r/,2+.j-V(»i2_„,2)-] 

 E=«3frti'+TV(ni'-?^2^i]+Oif«3'+TV(".'-"2-)] ^ (2) 



-"[(-'^'^)©HD(|)<i)(|)] 



This enables us to find the magnitude S of an area whose position only 

 is given, when the three other areas Sj, S2, S3, are given both in magni- 

 tude and position. 



Now let each of the four areas be divided by equidistant ordinates 

 into as many subdivisions as there are units in the bases tii^ iu^ ii^ and 

 n respectively, these bases being supposed to represent whole numbers, 

 and let cii, «3, and x be each a whole number or a whole number and a 

 half, according as ^h+^z, ^24-^3, and «2+w are respectively even or odd; 

 then all the subdivisions of the areas will be so situated that the ab- 

 scissas corresponding to their middle ordinates will be terms in an 

 arithmetical progression, and, consequently, the subdivisions themselves 

 Avill be terms in a series of the second order. We may regard these 

 subdivisions as representing not areas merely, but magnitudes of any 

 kind, and the areas Si, S2, S3, and S being the sums of groups of sub- 

 divisions, we see that formula (2) enables us to find the sum S of any 

 group of consecutive terms in a series of the second order when the 

 sums Si, S2, S3, of tlie terms in any other three groups in the s<M'ies are 

 given. From the sums of the terms in each group their aritliinetical 

 means are known, and vice versa, for n^, n-y, "3, and n are given, and these 

 are the numbers of terms which the several groups contain. The 

 groups may be entirely distinct, or they may overlap each other so that 

 some terms belong to two or more of them at once. The intervals be- 

 tween the middle point of the group S2, and the middle points of the 

 groups Si, S3, and S are «], ^3, awd x respectively ; the interval between 

 the middle points of any two consecutive terms being unity. We must 

 regard «i and ((3 as always positive, while x may be either positive or 

 negative. When n is made equal to unity, the formula gives the value 

 of a single term S by means of the suras Si, S2, S3, of the three given 

 groups of terms. The results are exact when the series taken is of the 

 second order, but if it follows some other law, or is irregular, approxi- 

 nmte or adjusted values for S will be obtained,«and if the same groups 



