288 METHODS OF INTERPOLATION. 



The origin of coordinates is the same as in the previous case, being at 

 the point of division between the third and fourth groups. When any 

 six consecutive terms have been computed and their five orders of dif- 

 ferences are taken, the rest of the series is easily constructed. It is 

 given in column (c). The sums of the terms for the decades 20-29, SO- 

 SO, &c., are the same as in the original series. 



It may seem strange that the two series (6) and (c) should differ so 

 much as they do, especially at the earlier ages. There are two reasons 

 for it. In the first place, they are derived, from two different sets of 

 groups J and as the original series is extremely irregular, the sums Si, 

 S2, &c., must vary somewhat from, their normal value, and vary differ- 

 ently in the two series, thus affecting the values of all the single terms. 

 This source of error, however, can be very much diminished, if not 

 entirely removed, by making a preliminary adjustment by the second 

 method, as will be shown hei-eafter. In the second place, there is an 

 essential difference in the nature of the two series (b) and (c). In {b) 

 the general term n is expressed l)y a polynomial of the fourth degree 

 in 00. When the two values -\-co and -co are assigned to x, the result- 

 ing value of u will have the same sign in both cases, because the 

 highest i^ower of x is an even one. But in the equation of series (c) the 

 highest power of x is odd, so that the values x=i=-\-co and x— — co will 

 give contrary signs to u. In general, when a series of an even order, 

 such as (h), is extended indefinitely in both directions, its terms will go 

 on increasing algebraically at both extremities, or diminishing at both 5 

 but a series of an odd order like (c) will increase at one extremity and 

 diminish at the other. It is evident that the original series (a) tends to 

 increase at both ends, as also does (/>), while (c) diminishes at the earliest 

 ages and increases at the latest ones. This has a considerable effect on 

 the form of the series. In (h) there is a minimum of .Gill? at the age 

 30, and no maximum at all, while (c) has its minimum of .72020 at the 

 age 34, and a maximum at 24. It appears that {h) represents (^0 more 

 faithfully than (c) does, and in like manner we may presume that in this 

 case a series of the sixth order would be better than one of the seventh 

 order, and, in general, that if a given series tends to increase at both 

 ends, as any rate of mortality of this nature does, or to diminish at 

 both ends, its representative series ought to be of an even order, while 

 if it tends to increase at one end and diminish at the other, the new 

 series should be made of an odd order. But there will be some excep- 

 tions to this rule, and of course, other things being equal, the greater ^ 

 the number of groups taken, and the higher the order of the new series, 

 the more faithfully will the original one be rei)resented by it. 



SECOND SrEXnOD OF ADJUST3EENT. 



If in formula (2) we make iii an odd number, and assume — 



