METHODS OF INTERPOLATION. 285 



In like manner, assuming seven groups, with a curve of the sixth degree, 

 we liud the seven constants — 



^= m520n ^^^^^^ S4+954(S,+S6)-7G31(S3+Ss)-75(Si+S;)]^ 



i«i' 

 |?i? 



B=j^^^,[9455(S5-S3) + 259(S,-S0-223G(So-S.)] 



C=3^;j^3[3435(S3+S5)+37(Si+S,)-6020Si-4G2(S,+ Se)] 

 ^=r7s^Io2{S,-B,)-S3{S,-&,)-7{S,-S,)] >(E) 



<j / 1> III 



r=24(J^/5(S5-S3)+(S,-Si)-4(S«-S2)] 



G=,=^,[15(S3+S5) + (Si+S,)-20S4-G(S2+S,)] 



So also with eight groups, and a curve of the seventh degree, the eight 

 constants are — 



A=3^^^^J11193(S4+S5)+609(S3+S,)-2919(S3+So)-63(Si+S,)] 

 B=5^^|^,[7175(S5-S4) + 119(S, — S2)-889(Sg-S3)-9(S3-S:)] 

 C=53^3[273(S3+So) + 7(Si+S3)-215(S,+S5)-G5(S,+S,)] 



r-jg^75(S5-S,) + ll(S,-S,)-41(S6-S3)-(Ss-Si)] 

 G=Xn^7[0(S3+SG) + (Si+Ss)-5(S4+S5)-5(S2 + S,)] 



In the same way we miglit determine the nine constants for a curve of 

 the eighth degree, and so on ; for the operations required, thougli some- 

 what tedious, are always possible.* We have found, then, a very simple 

 and general method by which, when any ?w+l consecutive groups of 

 equal extent are assumed in a given series, a new series of the mth 



* See formula (G) iu Ai^peudix I. 



