336 METHODS OF INTERPOLATION. 



used when the number of terms grouped together is odd, and the other 

 ■when it is even ; it regards the terms as being geometrically represented 

 by ordinates, instead of areas, and does not permit the use of groups 

 composed of a fractional number of terms, and it is not generally ai^pli- 

 cable to functions of other forms than those specified. 



APPENDIX IV. 



ADDITIONAL FORMULAS FOR INTERPOLATION WITH A CIRCULAR 



FUNCTION. 



Denoting by N the whole number of terms in the circular period, let 



2 r 



US Avrite '^ = 0: then assuming the curve — 



2/=A+if4Bi sin Or^) + Cicos {xO)\-\-^e[B2Hm2{xe)-\-C2COs2{xO)\ ) 



+ |^[B3sin3(.r<y) + C3Cos3(.r^)] + &c. y ^ 



we shall have for the sum of the terms in any group — 



S= Am + sin i («^y)[Bi sin {xo)-]-Gi cos (xo)] ^ 



+ sin ^ {nO)\B2 sin 2 {xO)-\-C2 cos 2 (xO)] \ (78) 



+ sin ^ {71 0)IB3 sin 3 {xO)+C3 cos 3 (*-<?)]4-&e. ) 



From this we can derive formulas for computing the values of the con- 

 staats A, Bi, Ci, B2, C2, &c., just as formulas (A), (B), (C), &c., were 

 derived from the algebraic formula (11) ; or, otherwise, we can determine 

 the constants by treating the equations of condition in the manner 

 peculiar to the method of least s(]nares. The results are the same in 

 either case. When the N terms are divided into three consecutive 

 groups of equal extent, we shall have — 



A=i(Si+S,+S3) ^ 



[a) 



Bi=f(S,-Si) 



Ci=-^- sin G0o[2S2-(Si+S3)] 



s 



With four groups, we get — 



A = ^(81+82+83+84) 



Bi=^[(S3-S2) + (S,-S,)])(*) 

 Ci=i[(S2+S3)-(Si+84)] 



B2 = i[(S3-S2)-(S4-Sl)] 



We omit the formulas for five, seven, nine, &c., groups, which are not 

 required in practice, the common use of monthly or hourly data in 



