O OQ 



METHODS OF INTERPOLATION. 



United States Coast Survey Eeport for 18G0, page 392. Coluuii) (1) of 

 the following table sbows the terms of the given series corresponding 

 to each hour of the day : 



It is required to represent this series by a formula containing five con- 

 stants. We will not make any ])reli!ninary adjustment by the second 

 method, as that is not indispensable to our system of interpolation by 

 groups, although it is generally desirable, as, indeed, it would be in a 

 less degree with Cauchy's method, Avhich also depends on the summa- 

 tion of irregular series of quantities within certain intervals. Dividing 

 our 24 given terms into six groups of equal extent, we get — 



S,= .45 • S,=.15 S,= -.73 



S.= — .57 S^=.G7 Sg= .05 



Computing by formula {c) the values of the first five constants, and 

 substituting them in (78), we have — 



S=.0008+ sin ^nO)[— .0667 sin (^^) + .1848 cos {xO)] 

 + sin {nO)[.:mu sin 2(.r ^y) + .7544 cos 2{x0)] 



which we transform into — 



S=.0008+.1965 sin ^{nO) sin (xO+W^^oV) 

 + .8144 sin {nd) sin {2x0+(u^o:y) 



This expresses the sum S of any group of n terms in the graduated 

 series, the abscissa of the middle point of the group being x, and each 

 term being supposed to occupy, on the axis of X, a space equal to 



2t 

 unity. The angle 6* is vm:=15o. If we further take «=1 and S=», we 



obtain the equation of the graduated series — 



«=.001 + .026 sin {xO-\-Um<^r>i')-\-.211 sin {2x0-\-61Oo3') 



Ymm this the values in column (2) are computed. The sums of the 

 terms in its six groups are, of course, not precisely ecpial to those in 

 column (1). To make them so,, it would be necessary to add to the 

 equation the term containing the sixth constant B3. This term is — 



+.018 sin 3(x0) 

 The origin of co-ordinates is at the middle of the series. If we wish to 



