K. Pearson 
299 
at once connect the ordinary constants s, c, g, k with my s, c', g', k'. Clearly k' is 
a nuniber of living persons like k, and s', c and g' are mere numerical quantities 
independent of what unit of duration of life we may select — day, month, year, etc. — 
while the s and c of the usual notation involve the unit in which life is measured — 
a theoretical, although scarcely practical disadvantage. Taking logarithms I now, 
dropping dashes, write the formula 
L = K+ Sj + Ge'^^^'^ (xxvii), 
where 
e^" = c' (xxviii). 
We must now proceed to find the area and first three moments A, Afi^, Afj,2, 
AfXs of (xxvii) about the middle of the range I. If we then equate these to the 
moments found from the table we shall have equations to determine K, S, G and n 
and therefore k, s, g and c in a perfectly direct and systematic manner, using all 
the data provided. We have 
Or if 
Ci^ Gl 
tto = A/l (xxix), 
„ sinh n 
^o = }<:+G—-- (xxx), 
Or if 
r+ii 
Afjbi= f Lxdx. 
J - hi 
a, = 12Aij.Jl^ (xxxi), 
o fcosh n sinh n") 
a, = S + 6G{ ~y (xxxii), 
f + il 
A/M.2 — j Lx'dx. 
Or if 
a^^ 12 A/x.,/l^ (xxxiii), 
„ ( sinh TO 2 cosh ?? 2 sinh ri) 
a^ = K+SG ] — + — [ (xxxiv), 
Or if 
Afx,3= i Lx^ dx. 
J -u 
-ii 
a, = 80Afi,/l' (xxxv), 
a , io/-r(cosh9i 3 sinh n 6 cosh n 6sinh«] 
a, = S + lOG \ — + :~ — \ (xxxvi). 
From (xxxii) and (xxxvi) we have : 
,^(4cosh?i 24 sinh » 60 cosh ?i 60 sinh ?i] , 
2 ^ \ ~x — ^ ...(xxxvii). 
a-A - a, 
