METHODS OF INTERPOLATIOX. 305 



assign to J7 tbe values — i, +i, +^, &o., in snccession, the resulting 

 values of u will be the logarithms of the probabilities of living one year 

 at the ages m-\-5^j ?«+<>A, ^" + ''^^5 &c- If we take .r=0, tlie value of 

 u will be the logarithm of the probability of living one year at the age 

 m+6, and w^e shall have the simple formula — 



logi^n, +G= 3^7-1 S,—33(S, + S3)] . . . (34) 



To illustrate the use of this by an example, and to test its accuracy at 

 the same time, let us suppose that there is no migration, and assume 

 that, in accordance with the English Life Table, No. 3, for males, the 

 population living at the first census, between the ages of 54 and 55, 55 

 and 56, 50 and 57, respectively is — 



P54=2120G1, P55=20C984, P56=201772 



and that the survivors at the second census are — 



r6, = 154139, Pc5 = 147319, Pee = 140299 



The logarithms of the probabilities of living ten years at the three ages 

 54^, 55i, and 5Gi are therefore — 



Si = log Pe4 — log P54 = 1.8014518 



S2=logPe5— logP55=-1.8523219 



S3=log Pee— log Pse = 1.8421937 



and since ?«=54, we find that the logarithm of the probability of living 

 one year at the age ;» + 0=C0 is — 



log j;6o=8V[74 S2-33(Si+ S3)] =1.9850440 



This value differs but very little from the one which is actually given 

 by the English table, namely — 



logi>Go=log/ei-log7co=log 170421 -log 182350=1.9850445 



The method followed in the above example will be found sufficient for 

 the determination of the probability of living one year after every birth- 

 day, except the first nine or ten of childhood and the last seven of old 

 age. With the help of formula (33) we can find the probabilities for all the 

 ages of childhood, except the first three or four, by assigning to a' the nega- 

 tive values — 1, — 2, — 3, &c., which will give values for log2)m+5, 

 logi>m^-4, losihni.3, &c. So, too, for the last years of life, w^e can find 

 log Pn, + 7, logj^u, + 8, \og2hn + 9, &c., by assigning to x the positive values 1, 2, 

 3, &c. This will comi)lete the series of values of log p from early child- 

 hood to extreme old age. As it will be alreadj^ approximately adjusted, 

 nothing more will remain but to divide it into groups of an equal num- 

 ber of terms each, and to make the final graduation by either the first 

 or the third method. There will be a convenience in graduating the 

 logarithms instead of the corresi>onding numbers, because logj>, and 

 not p itself, is what we require for computing in the most expeditious 

 manner the numbers living to attain each year of age out of a given. 

 20 s 71 



