558 
ME. B. GOMPEETZ ON THE SCIENCE 
By direct calculation . . '000887, '000537, Tc^'^=. '000350, 
By the anticipating formula k&'^— '00106, kz^^^— '00129, kz^"^— '00355, k^}^— 
W„=- -00018, W,,= - -00075, Wi3=- -00320, 
where x is not divisible by 5 ; but in cases where x is divisible by 5, W,^ is always equal 
to 0, and among the cases the three negative values are 
--00018'! 
--00075l=--00435. 
--00320j 
Their sum differs very little from the one positive case -00375, and they are all per- 
fectly insignificant compared to the total values of XLji = 3-808 &c., ?iLi 2 = 3-806 &c., 
>,Li 3 = 3-804 »&c., >iLi 4 =: 3-803 &c. ; but when I say perfectly msignificant, I mean uith 
respect to the chance of living, — though that difference would, in a small degree, cause 
the number answering, say to the age of 11, to make it apply to an age very triflingly 
differing from the exact age of 11, but this is not of the slightest importance -uith 
respect to valuing chances of anticipation. It may be observed that when kz"" is very 
small, as in the case above enumerated, we see that the anticipated values of kz"" may 
have not only a large proportion to each other, but may ev^en be of contrary signs, as kf 
cannot be negative, though by the anticipating formula in the case of k it comes out 
negative ; but these are cases where the value is of no importance in consequence of 
its smallness; and for long before the age of 20, the effect of kz"" in the anticipation can 
be omitted. 
And now with respect to the formula which is not as imperatively required as the 
others 
iU/y''+^=(when o:=60) 'Jo-f 
let the anticipation also be successively for 11, 12, 13, 14 years. I purposely take 
years not divisible by 5, because, as I have said above, when ^’=5, and a-\-x does not 
exceed 100, 'W^, is invariably =0, the variability of /a not coming into play; we have 
The real values of p"' = -0013630, ^^^=-0017573, p73^.oo22656, -0029217 
Anticipating formula = -0013264, -0017106, -0022601, '0029430 
'W7,=-0000366,'W-2=:-0000467,W-3=-0000055,W.4= -'0000213 
The mode above described of getting rid of the annoyance of the non-convergency of 
the developed expression /rs* is particularly worthy of attention, and is entitled to be 
noticed by a name which I call the Interpolation by selected terms taken ^er saltum, as 
it offers another and an easier mode of calculating the three analytical anticipating 
Tables mentioned, which are so efficient in practical valuations; and calculatmg those 
Tables independent of each other when, as in the mode pointed out first, required the 
calculation of the value of and from that the value of and lastly, the value 
