312 METHODS OF INTERPOLATION. 



Nevertheless, such trials as have beeu made with this system of group- 

 ing have not resulted favorably for its use in constructing mortality 

 tables. The series seems to be rather distorted by it. This is shown 

 when we construct by formula (42) a series of the fourth order to repre- 

 sent the given series (/). Here we have iSr=90, and consequently — 



7)i=n,=S.5M235, ii,=n,=22^, 7?3=27.81153 



so that the suras of the terms in the five groups, as found by the aid of 

 formula (39), are — 



Si= 3.G3932 83= 68.3G19 



82=17.60021 84=337.0553 



85=297.960 

 the five constants are found to be — 



A= 1.919514 € = .008277894 



B= .1673728 D = .0001512150 



E=.0000006G35611 

 and the equation of the graduated series stands — 



^=1.920204+ ,1674106 x+ .008278226 x^-\- .0001512150 x^ 

 + .0000006635611 a;* 



If the values — J, +^, +f, &c., are assigned to x, the resulting values 

 of w are the terms in the graduated series for the ages 54, 55, 5G, &c. The 

 sum of all the terms in the series is equal to the sum of all the terms in 

 (/), as it should be. But it does not afford a good representation of (/), 

 especially in the first half. It begins at the age 10 with the value 

 .14024, goes on increasing up to the age 27, where it has a maxiuuim 

 of .81152, then diminishes up to the age 36, where it has a mini- 

 mum of .77662, then increases to the close, having the value 41.690 at 

 the age 99. 



On the other hand, if we construct by formula (C) the equation of a 

 simihn- series from five consecutive groups of eighteen terms each, the 

 sums of the terms in the groups are — 



81= 9.82520 83= 39.94320 



8, =16.89333 84=154.96600 



85=502,98900 

 the five constants are — 



A=2.023103 €=.007188222 



B= .1433032 D=.0001722763 



E =.000001434104 



and the equation of the graduated series is — 



« =2.023702+ .1433463 x + .007188939 .t^+ .0001722763 x^ 

 + .000001434104 x-^ 



This represents (/) with a considerable approach to accuracy, commenc- 

 ing at the age 10 with the value .32319, increasing continuously tliere- 

 after, and terminating at the age 99 with the value 43.443. This exam- 



