METHODS OF INTERPOLATION. 331 



three roots of the equation of relation must in all cases be positive ; if 

 any of them are negative, the inference will be tliat the given series 

 cannot, for purposes of interijolation, be represented by an equation of 

 the proposed form. 



If the number of constants is odd, for instance, seven, we shall find the 

 Bcale of relation from the four equations — 



Ao(So-A') + Ai(Si-AO + Ao(S2-A') + (S3-AO=0 



Ao(Si-A')+Ai(S3-AO+A2(S3-A') + (S,-A')=0 



Ao(S.-A') + Ai(S3-A') + A2(S4-A') + (S3-AO=0 



Ao(S3-A') + Ai(S4-A') + A,(S5-A') + (Sc-AO=0 



first eliminating- A' by subtracting each equation from the succeeding 

 one. The equation of relation will be of the same degree as in the pre- 

 vious case, and the values of A', B', C, and D' will be found from the 

 four equations of condition — 



8o=A'+B'-fC'+D' 



Si=A'+B'/3''+CV''+B'r?* 



S2=A'+B';'j2''-f-C'r"+D'o2* 

 S3=A'+B'/53''+CV"'+D'o3/. 



If the number of coustants and of groups assumed were eight or nine, 

 the mode of procedure would be precisely similar to the above. The 

 scale of relation would contain four terms, and the four roots of the 

 equation of relation — 



c^+ A3 5^+ A2 z^'+Ay cr+ Ao=0 

 would be the values of the four constants /J*, 7-", o", e*. 

 In the simiilest case of all, we have the curve — 



whose equidistant ordinates are in geometrical progression. If we 

 assume — 



y=A+(Blog'/^)/3' 



it is easy to obtain the following: 



0__ / S2— S A A 



Si — So 



B= 



(07) 



S=Ah+b(,5*"-9-*"^,3^ 

 This can often be used with advantage in i)lace of (3) or any similar 



