COimECTED WITH HUMAN MOETALITT. 
555 
There seems to be a distinction between the law of mortality and that of sickness, 
inasmuch as in the law of mortality, is a positive fraction less than unity, whereas 
the similar term B in the law of sickness is greater than unity. 
Art. 29. Here I must draw the reader’s very particular attention to the formula 
where 
where for analytical anticipation it is put in the form 
and the two portions and are developed into series proceeding by the powers 
of X, because those series for several of the first terms, when x is at all large, are so 
divergent that in that case they become of no practical service ; and in fact, though, if a 
sufficient number of terms be used, they are ultimately convergent and would lead to 
the true value, they present so formidable an obstacle as to cause, if those series be used, 
a complete refusal of the aid which was expected from them to render the formula 
analytically anticipatory, at least at those periods when these functions have a prevailing- 
influence over anticipation ; and if this difiiculty had not been overcome by the adoption 
of a subterfuge, which I almost despaired of finding, a great part of the highly im- 
portant analytically anticipating powers of the formula would be destroyed. The series 
alluded to are exhibited as follows : — 
and 
" — ~1“ £>^4" 2 ^1 3 ' &C.^, 
/ 1 --^ 1 \ 
— _ j,| ^3^ &C. j. 
And alluding to the first of these, I observe that in the formula above, when we refer it 
to the Carlisle mortality, ^£=1-79811; ^£= — -46427; _=^=T'16855 ; ^=-14742: if we 
wished to anticipate only for ten years forward, zx would be, say about — and the series 
l+£,r+-|£| &c. Avould be 
9 
1 
9 
1^^ ^ 
2 
2.3 
2 
' 2.3.4 ‘2 
„ _T 999 999,9999 „ 
1 2'^2’4 2‘4'6‘^2’4’6'8 ’ 
SO that we should not only have to go to the flfth power of x before the series began to 
converge, but to go to much higher powers, whereas the real value of sought is but of 
the small value of about -001411, of very small value, which may be considered even of 
insignificant value with respect to the total value of notwithstanding its perplexing 
annoyance. But if we wished to anticipate for twenty years, although if a is even equal 
to 0, the value of the function is of such total insignificance with respect to the 
other portion of being no greater than about *000014, whilst the value of 
3-81023, yet in the development of into the series above, the number of 
diverging terms alternately positive and negative of greater value before the series even 
