To illus

215. The Mixing of Pure Forms. Heredity. trate this fundamental fact, let us consider the history of two characters brought together for the first time, and the manner in which they will naturally appear in the offspring.

To put the matter in its simplest form, let us suppose a herd of pure black cattle to meet and mingle with a herd (of equal numbers) of pure red, and that they mate without restraint,that is, without selection. This being true, one half the offspring of the black females will be pure black (designated by B2) and one half will be mixed, black and red (designated by BR). The same principle applies to the red females, whose progeny will be equally divided between the mixed offspring and the pure reds. Expressed in tabular form we should then have:

For every 200 offspring of black females: 100B2+ 100BR
For every 200 offspring of red females:

Total distribution, 400 offspring:
The proportions are:

100BR100R2,

100B2+ 200BR + 100R2, B2+ 2BR + R2.

It is evident that, whatever the numbers involved, the above is the proportion in which the pure and the mixed forms will naturally appear in the first generation of admixture between two pure forms. From this we see that indiscriminate breeding of distinct characters results in both pure and crossed (or mixed) forms in their descendants, and this is in the proportion of 121 in the first generation.

216. The Second Generation. What now will be the character of the next generation, as bred from the individuals B2 (pure black), BR and BR (mixed), and R2 (pure red)?

Continuing the assumption of indiscriminate mating and uniform fertility, and remembering the relative numbers involved, we shall have the following:

It is to be noticed that this total is a complete expression of the binomial B + R expanded to the fourth power according to the binomial theorem.

217. Successive Generations. Succeeding populations follow the law of the binomial theorem except as interrupted by selection or differences in fertility. In the breeding of this third generation among themselves, we find the numbers becoming rapidly complicated; but from the fact that it always follows the binomial theorem, we can write the normal distribution for the fourth generation of descendants of any pair of characters as follows:

B8 + 8B'R + 28B6R2 + 56B5R3 + 70B4R1

+56B3R5 + 28B2R6 + 8BR2 + R3.

218. Combinations of Three Characters. Though the numbers become rapidly more complicated, the same principles apply when dealing with three or more characters. For example, suppose we introduce a third color, white (W). We shall then have as the result of the first mating the following:

(B+R+W)2

=

B2 + 2BR + R2 + 2BW + 2RW + W2.

In case the number of R animals is twice the number of B animals we shall have as the result of the first mating the following distribution:

(B + 2R)2 = B2 + 4BR + 4R2.

EXERCISES

1. Plot a few graphs using binomial coefficients as ordinates and the number of the corresponding term as abscissas.

2. How many varieties of sweet peas are produced by sowing in the same bed three different strains (a) first year; (b) second year.

Ans. (a) 6; (b) 14.

3. A farmer buys two different kinds of thoroughbred chickens but allows them to mix freely. How many different kinds of chickens will he have at the end of (a) the first, (b) the second, (c) the third year of hatching? Ans. (a) 3, (b) 5, (c) 9.

4. Four different varieties of wheat are planted side by side. How many different varieties will be harvested?

Ans. 10.

5. Plot graphs as indicated in Ex. 1 for the results of Ex. 3. 6. What varieties and in what proportion are obtained by freely mixing the first and second generations?

7. I plant 8 sweet pea seeds-4 red, 4 white. Each seed produces 16 flowers each flower matures 2 seeds which germinate and grow the following season. Find the total number of flowers, the proportion and number of the different kinds of flowers, in the (a) first, (b) second, and (c) third generations.

CHAPTER XVII

THE COMPOUND INTEREST LAW

219. Compound Interest. If a dollars are loaned at r per cent. per annum, compound interest,

If the interest is compounded semi-annually instead of annually the amount at the end of x years is

if compounded monthly the amount at the end of the same period is

and if compounded n times a year the amount is

a [ 1 + 100n]"

220. Continuous Compounding. The Compound Interest Law. If we find the value of this expression as n increases indefinitely, we find the amount if the interest were compounded

continuously. For convenience let r/100n be represented by 1/u.

As n increases indefinitely, u increases indefinitely. As u increases indefinitely the expression (1 + 1/u)" approaches a definite limiting value. The right member of equation (2), as u increases indefinitely approaches a definite limiting value. In as much as the two members of equation (2) are equal for any finite values of u, they approach the same limiting value. We have then

The sum on the right hand side is represented by the symbol e and is approximately equal to 2.7182818. This number e is the base of the Napierian system of logarithms. The sum of the first seven terms of the series (3) is 2.7180555.

the first eleven terms gives 2.7182818.

*lim A means the limit of A as u increases indefinitely.

The sum of