CONJSTECTED WITH HUMAN MOETALITY. 
557 
There may be cases, indeed, when we should not stop ; but though we have not data 
for ages beyond 100, the coefficient in the expression though taken as suffi- 
ciently expressed by the constant value till a-\-x does exceed 100, is evidently not abso- 
lutely constant ; for if so, a person would be represented to be capable to live for ever ; 
and I therefore satisfy myself by omitting to introduce a term for which I have no data, 
by limiting my term of anticipation to the period when a-j-s becomes =100. 
I now observe that for the Carlisle mortality (where X|M/=il’29927, Xj'=T10349), 
iM/v®®'''^® give the following series of terms: — 
@83303, @10572, @37662, 013417. 
And adopting 
^J,60 + * 
and putting 
the same mode of interpretation as in the former case, and representing 
'T — 'T 
J-O tl 05 
To+T,a:-l-T2^"^-T3^'+'I4^^ 
I'T 'T 1_'T 'T L.'T 'T 
g J-1 ^15 52 '*-2 ^2? 53 '*“3 *^31 
49 
we have 
To='000083303, '!,=- -00040315, T2=+-00121095, 'l3=--00065977, T4= + -0001489, 
^_'j^=5.98816, -^'12=5-68517, -=—'13=6-72248, -='14=6-23825, 
'lo=-000083303, 'l4=-@97313, 'l2=@484363, 'l3=-@52753, 'l4=@23825. 
But here it is necessary to observe, that though these series for finding are 
exactly true with respect to any value of if it be divisible by 5 (limited with respect 
to to the case observed above, a-\-x not being greater than 100), still they are not 
identical with that expression, as they may differ both in plus and in minus with them, 
if X, when divided by 5, should leave a remainder, say of 1, 2, 3, 4, which are four cases 
of more exact identity ; but as the variance from identity is quite insignificant with 
respect to the value of <=-L^+j„ as will be shown further on, it is perfectly allowable to 
use this method without paying the slightest attention to the more absolute identity of 
the expression ; but to proceed to prove this assertion. I represent by 
IJ+l V+laO?®-!- 
and also represent jw-v®®'*'® by 
JoH" ll>^ + 'l2^^"l“'l3^^-4"'l4^^“l“'W^, 
and I am to show that W^, and 'W^ are either equal to nothing, or are insignificant 
with respect to whether they shall turn out to be positive or negative. Now I 
observe in the case where x is divisible exactly by 5, the above investigation shows that 
and 'W^ are both absolutely equal 0 ; and for the rest I will take x, by way of 
example, successively 11, 12, 13, 14, in which x divided by 5 leaves either 1, 2, 3, or 4, 
and where the anticipation is respectively for 11, 12, 13, 14 years. 
