246 NATURAL INHERITANCE.

(a+bx,4 i.e. suppose that to= (a+b) ° ti -qua-11 b)Q and so on.

Herefi (x)J(+bb Q and Imo =(a+b)Q

f (x) - 1 1 a + b(a + bx)' ~ q (a + b)Q (a+ b)Q 2"40 (a + b)Q { a + blmo 1 4

Generally .mo = (a + by { a + b„,mo } q = (a + b) { b + *-imo } q Q

If, therefore, we wish to find the number of extinctions in any generation, we have only to take the number in the preceding generation, add it to the constant fraction a , raise thee sum to the

b

?

power of q, and multiply by b (a+b)Q

With the aid of a table of logarithms, all this may be effected for a. great number of "generations in a very few minutes. It is by no means unlikely that when the true statistical data to, t1, etc., t, are ascertained, values of a, b, and q may be found, which shall render the terms of the expansion (a + bx)Q approximately proportionate to the terms off, (x). If- this can be done, we may approximate to the determination of the rapidity of extinction with very great ease, for any number of generations, however great.

For example, it does not seem very unlikely that the value of q

might be 5, while to, t1...tq might be •237, •396, -264, •088, •014, •001, or nearly, , , i 4, , i'u-9, and 1 u~ u.

Should that be the case, we have, fl (x) = (3 +x)5 Imo = 35 and generally *mo = 45 { 3 +,.1mo } s

Thus we easily get for the number of extinctions in the first ten generations respectively.

237, •346, •410, •450, •477, -496,,510,,520, •527, •533.

We observe the same law noticed above in the case of 1 +x+x2 3

viz., that while 237 names out of a thousand disappear in the first