METHODS OF INTERPOLATION. 339 



transfer it to the first term, we put u:-— 11^ iu the place of x, and thus 



u= .001+ .026 sin {x0+297O21')-\- .211 sin {2x0-^82053') 

 wliich does not differ greatly from the equations obtained by (.'auchy's 

 metliod and the method of least squares, as given in tlie Coast Survey 

 Report. 



Similar results would be obtained by dividing the given series into 

 eight or twelve groups, and computing the values of the first five con- 

 stants from fornuilas (d) or (e). These results would probably be a little 

 more accurate than the preceding, being in accordance with the prin- 

 ciple of least squares, as already stated. 



In cases where the data for interpolation are the mean values Mi, M.,, 

 M3, «&c., of the ordinate, taken within intervals formed by equal divisions 

 of the circular i)eriod N, our formulae {a), (/>), (c), &c., will still be ap- 

 plicable. For instance, with three intervals, we shall have — 

 S,=pI,N, S,=pI,N, S;,=iM3X 



Formula (a) then gives the values of the three constants, and since 

 S=M«, formula (78) becomes — 



M=A+- sin ^(n())[B^ sin (.r^)-f C, cos (xO)] 



it " 



which expresses the mean value M of the ordinate within any interval n. 

 To illustrate this, let us take the corrected mean temperatures at New 

 Haven (Transactions of the Connecticut Academy, Vol. I, p. 233) for 

 intervals of four months : 



January to April M,=3io. 35 Fahr. 



May to August 3L=.G6o. 84 



September to December Mj^^iOo. 15 



To obtain from these an ecjuation for the series of daily means, we have 

 ]Sr=3G5i, and consequently' — 



Si =4182, S,=8138, S,=5019 



Formula («) then gives — 



A==49.11, Bi=958, Ci=2492 



and (78) gives — 



M=49.1l4-2G7(!f ^|-^ sin ^nO) sin (.t'^^+OSOoS') 



This equation expresses the mean temperature of any interval of n 

 days. The angle is — 



If wc also take n=l, the equation of daily means is found to be — 



M=49. 11+22. 91 sin (.r ^y+08058') 

 The origin of co-ordinates is at the middle of the year. 



