chengyewang/TexOCR-eval huggingface dataset:
respectively, then the relations between the components $u_1, u_2, u_3$ with respect to the $x$-system and the components $\omega_1(\xi_1, \xi_2, \xi_3)$, $\omega_2(\xi_1, \xi_2, \xi_3)$, $\omega_3(\xi_1, \xi_2, \xi_3)$ with respect to the new $\xi$-system are given by the equations of transformation

\[
\omega_1 = a_1 u_1 + \beta_1 u_2 + \gamma_1 u_3
\]
\[
\omega_2 = a_2 u_1 + \beta_2 u_2 + \gamma_2 u_3
\]
\[
\omega_3 = a_3 u_1 + \beta_3 u_2 + \gamma_3 u_3
\]

and

\[
u_1 = a_1 \omega_1 + a_2 \omega_2 + a_3 \omega_3
\]
\[
u_2 = \beta_1 \omega_1 + \beta_2 \omega_2 + \beta_3 \omega_3
\]
\[
u_3 = \gamma_1 \omega_1 + \gamma_2 \omega_2 + \gamma_3 \omega_3
\]

respectively. (Cf. Chap. I, p. 6.) The components $\omega_1, \omega_2, \omega_3$ in the new system thus arise from the introduction of the new variables and the simultaneous transformation of the functions representing the components in the old system.

When in physical applications each point of a portion of space has assigned to it a definite value of a function $u = f(x_1, x_2, x_3)$, such as the density at the point, and we wish to emphasize that the property is not a component of a vector, but on the contrary is a property which retains the same value although the coordinate system is altered, we say that the function is a scalar function or scalar, or, if we wish to emphasize the association between the values of the function and the points of the portion of space, we speak of a scalar field. Thus for every vector field $u$ the quantity $|u|^2 = u_1^2 + u_2^2 + u_3^2$ is a scalar, for it represents the square of the length of the vector and therefore retains the same value independently of the coordinate system to which the components of the vector are referred.

In the examples above the vector field $u$ is given us to begin with, and its components with respect to any system of rectangular coordinates are therefore determined. If, conversely, in a definite co-ordinate system, say an $x$-system, there are given three functions $u_1(x_1, x_2, x_3)$, $u_2(x_1, x_2, x_3)$, $u_3(x_1, x_2, x_3)$, these three functions define a vector field with respect to that system, the components of the field being given by the three functions. To obtain the expressions for the components $\omega_1, \omega_2, \omega_3$ in any other system we have only to apply the equations of transformation deduced above.
Quantities of one kind are said to be inversely proportional to quantities of another kind, if the ratio of any two of the first kind is equal to the inverse ratio of the corresponding two of the other kind.

If 6 men can do a piece of work in 4 days, then 8 men can do it in 3 days, or $6:8$ equals the inverse ratio of $4:3$, i.e. $3:4$. Hence the number of men required to do some work, and the time necessary to do it, are inversely proportional.

\textbf{181.} In any proportion the product of the means is equal to the product of the extremes.

Let
\[
a : b = c : d,
\]

or
\[
\frac{a}{b} = \frac{c}{d}.
\]

Clearing of fractions,
\[
ad = bc.
\]

\textbf{182.} The mean proportional between two numbers is equal to the square root of their product.

Let the proportion be
\[
a : b = b : c.
\]

Then
\[
b^2 = ac. \quad (\S\ 181.)
\]

Hence
\[
b = \sqrt{ac}.
\]

\textbf{183.} If the product of two numbers is equal to the product of two other numbers, either pair may be made the means, and the other pair the extremes, of a proportion. (Converse of \S\ 181.)

If $mn = pq$, and we divide both members by $nq$, we have
\[
\frac{m}{q} = \frac{p}{n}.
\]

\textbf{Ex. 1.} Find $x$, if $6 : x = 12 : 7$.

\[
12x = 42. \quad (\S\ 181.)
\]

Hence
\[
x = 3\frac{1}{2}.
\]

\textbf{Ex. 2.} Determine whether the following proportion is correct or not:

\[
8 : 5 = 7 : 4\frac{3}{8}.
\]

\[
8 \times 4\frac{3}{8} = 35, \quad \text{and} \quad 5 \times 7 = 35;
\]

hence the proportion is correct.
If the value of $\sqrt{9}$ had not been restricted (§255) to its positive value, 9 would be the root. On the other hand, no positive value of $\sqrt{x}$ can satisfy the equation, since 5 plus a positive number cannot equal 2.

Consequently equation (1) has no root, although the student should bear in mind that this is the consequence of the arbitrary exclusion of negative values of roots.

280. If $x = 9$ is not a root of the equation $5 + \sqrt{x} = 2$, it becomes necessary to ascertain why the solution produced this value.

In solving an equation, we usually proceed as if the given equation were true and we had to prove the correctness of each following one; or we prove that (3) is true if (1) is true. The real problem, however, is the opposite one, viz. to prove equation (1) is true if equation (3) is true. That is, we should start from equation (3) and prove successively (2) and (1).

But the members of (2) are the square roots of the members of (3), and if two quantities are equal, their square roots are not necessarily equal, as shown by the following illustration: $(-3)^2 = (+3)^2$, while $-3$ does not equal $+3$. Hence (2) does not need to be true, if (3) is true.

In general, squaring the two members of an equation introduces a new root, as can be seen from the following example:

Let

$x = a$.

Squaring both members, $x^2 = a^2$.

The roots of the second equation are $+a$ and $-a$, while the first one has only one root, $+a$.

281. Equivalent equations are equations which have the same roots; as $x + 4 = \sqrt{x}$ and $x = \sqrt{x} - 4$.

282. A new root which is introduced by performing the same operations on both members of an equation is called an extraneous root.
Find the arithmetic means between:

6. $a$ and $5a$.

8. $\dfrac{x^2 - y^2}{x}$ and $\dfrac{x^2 + y^2}{x}$.

7. $\dfrac{1}{a}$ and $\dfrac{1}{b}$.

9. $\dfrac{a-b}{2}$ and $\dfrac{3a+b}{2}$.

Find the series in which:

10. The 5th term is 19, and the 8th term is 31.

11. The 9th term is 12, and the 17th term is 60.

12. The 4th term is $-1$, and the 9th term is $-2$.

13. The 12th term of an A.P. is 14, and the 20th term is $-6$. Find the 28th term.

14. The sum of three numbers in A.P. is 18, and their product is 192. Find the numbers.

Hint. Let $x-y$, $x$, $x+y$ represent the numbers.

15. The sum of three numbers in A.P. is 9, and the sum of their squares is 29. What are the numbers?

16. The sum of five numbers in A.P. is 25, and the sum of their squares is 135. Find the numbers.

17. The sum of four numbers in A.P. is 20, and their product is 384. Find the numbers.

Hint. Let $x-3y$, $x-y$, $x+y$, $x+3y$ represent the A.P.

18. How many times does a common clock strike in 12 hours?

19. A bookkeeper receives every year an increase of \$100. How many dollars did he receive in 15 years if he received \$1200 during the first year?

20. For boring a well 200 yards deep a contractor receives \$1.00 for the first yard, and for each yard thereafter 1\cent more than for the preceding one. How much does he receive all together?
\section{Variables and Limits}

400. Functions (\S 292) are usually denoted by symbols of the form $f(x)$, $P(x)$, $F(x)$, etc., and are read $f$ function of $x$, $P$ function of $x$, etc.

Thus, $f(x)$ comprises all expressions which involve the letter $x$, and it may represent in one discussion $3x^2 + 2x + 1$, in another $\sqrt{x}$, in a third $4^x + 2x^3$, etc.

401. If $f(x)$ is known in any particular discussion, $f(a)$ is formed by substituting $a$ in place of $x$.

E.g. If
\[
f(x) = 2x^2 + 2x + 3,
\]
then
\[
f(3) = 2 \cdot 3^2 + 2 \cdot 3 + 3 = 27,
\]
\[
f(a) = 2a^2 + 2a + 3,
\]
\[
f(0) = 0 + 0 + 3 = 3, \text{ etc.}
\]

402. If $y = f(x)$, $x$ is called the independent variable, and $y$ the dependent variable.

403. Similarly, $f(x, y)$ denotes a function of two independent variables. Thus, $x^2 + 3xy + y^2$, or $\sqrt{x} + \sqrt{y}$, or $\frac{x}{y}$, may be represented by the symbol $f(x, y)$.

E.g. If
\[
f(x, y) = x^y + x,
\]
\[
f(2, 3) = 2^3 + 2 = 10.
\]

\subsection*{Exercise 142}

\begin{enumerate}
\item If $f(x) = x^2 + 3x + 2$, find $f(1)$, $f(0)$, $f(-1)$.
\item If $f(x) = 5x^3 - 2x - 3$, find $f(2)$, $f(a)$, $f(0)$.
\item If $f(x) = 4^x$, find $f(0)$, $f(-1)$, $f\left(\tfrac{1}{2}\right)$.
\end{enumerate}
438. Complex numbers. If $a$ and $b$ are real numbers, the complex number $a+bi$ may be represented by $OB$, the sum of $a$ and $bi$. I.e. Draw $OA=a$, and $AB$ equal and parallel to $bi$. $OB$ represents $a+bi$.

E.g. $OE$ represents $-4-2i$.

\begin{figure}
\centering
\includegraphics{figure_1.pdf}
\label{figure_1}
\end{figure}

439. The absolute value or modulus of any number (i.e. real, pure imaginary, and complex) is the length of the line which represents the number. It is always taken as positive.

The absolute value of $a+bi = OB$, or $+\sqrt{a^2+b^2}$.

The absolute value of $-4-2i = \sqrt{4^2+2^2} = 2\sqrt{5}$.

440. The amplitude of $OB$ is the angle $XOB$, i.e. the angle between $OX$ and $OB$, measured from $OX$ counter-clockwise.

Ex. 1. Determine the algebraic meaning of the rotation of a line through an angle of $60^\circ$.

Let

$OA = OB = OC = OD = 1$,

and

$\angle AOB = \angle BOC = \angle COD = 60^\circ$.

\begin{figure}
\centering
\includegraphics{figure_2.pdf}
\label{figure_2}
\end{figure}

If $x$ is the number which, applied as a factor, produces the required rotation,

then

$OB = x, \quad OC = x^2, \quad OD = x^3$.

I.e.

$x^3 = -1$,

or

$x = \sqrt[3]{-1}$.

The rotation through an angle of $60^\circ$ represents therefore a multiplication by $\sqrt[3]{-1}$, and line $OB$ represents $\sqrt[3]{-1}$. A simple geometrical deduction shows that $OB$ or $\sqrt[3]{-1} = \frac{1}{2} + \frac{\sqrt{3}}{2} i$.
489. If each element of a column is multiplied by the same number, and the products be added to (or subtracted from) the corresponding elements of another column, the determinant is not altered.

According to the preceding paragraph we have
\[
\begin{vmatrix}
a_1 \pm m b_1 & b_1 & c_1 \\
a_2 \pm m b_2 & b_2 & c_2 \\
a_3 \pm m b_3 & b_3 & c_3
\end{vmatrix}
=
\begin{vmatrix}
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2 \\
a_3 & b_3 & c_3
\end{vmatrix}
\pm
\begin{vmatrix}
m b_1 & b_1 & c_1 \\
m b_2 & b_2 & c_2 \\
m b_3 & b_3 & c_3
\end{vmatrix}.
\]

But the last determinant equals
\[
m
\begin{vmatrix}
b_1 & b_1 & c_1 \\
b_2 & b_2 & c_2 \\
b_3 & b_3 & c_3
\end{vmatrix}
\]
and hence vanishes (\S 485).

Ex. Evaluate
\[
\begin{vmatrix}
4244 & 4245 \\
4246 & 4247
\end{vmatrix}.
\]

The determinant
\[
=
\begin{vmatrix}
4244 & 4245 \\
2 & 2
\end{vmatrix}
=
\begin{vmatrix}
4244 & 1 \\
2 & 0
\end{vmatrix}
=-2.
\]

490. Evaluation of determinants. By means of the preceding proposition all elements but one in a column (or row) can be made equal to zero, and hence the determinant can be reduced to one of the next lower order (\S 480). In many cases, however, the determinant, before reduction to a lower order, should be simplified as follows:

(1) Remove factors common to all elements in a row or a column.

(2) Diminish the absolute values of the elements by subtracting the corresponding elements of other columns (or rows) or multiples of these elements.
Ex. 1. Prove that $5$ is a root of the equation

$x^3 - 3x^2 - 18x + 40 = 0.$

Dividing by $x - 5$,

\[
\begin{array}{r|rrrr}
5 & 1 & -3 & -18 & 40 \\
  &   & 5 & 10 & -40 \\
\hline
  & 1 & 2 & -8 & 0
\end{array}
\]

Since there is no remainder, $5$ is a root of the given equation.

507. If $f(x)$ is divided by $x-a$, the remainder of the division is equal to $f(a)$. (Remainder Theorem, Chapter XVI.) Using the notation of \S 505, we have

\[
(x-a)Q(x) + R = f(x).
\]

Substituting $x=a$,

\[
R = f(a).
\]

Ex. 2. If $f(x) = x^4 - 2x^3 - 9x^2 + 2$, find $f(4)$.

Dividing $f(x)$ by $x-4$,

\[
\begin{array}{r|rrrrr}
4 & 1 & -2 & -9 & 0 & 2 \\
  &   & 4 & 8 & -4 & -16 \\
\hline
  & 1 & 2 & -1 & -4 & -14
\end{array}
\]

Hence $f(4) = -14$.

508. The preceding method of substitution may also be demonstrated as follows:

\[
\begin{array}{r}
x^4 - 2x^3 - 9x^2 + 0\cdot x + 2 \\
+\;4x^3 + 8x^2 - 4x - 16 \\
\hline
x^4 + 2x^3 - x^2 - 4x \; ; \; -14
\end{array}
\]

I.e. if $x=4$,

$x^4 = 4x^3$, which added to the next term gives $+2x^3$.

$2x^3 = 8x^2$, which added to the next term gives $-x^2$.

$-x^2 = -4x$, which added to the next term gives $-4x$.

$-4x = -16$, which added to the next term gives $-14$.

Therefore $f(4) = -14$.
12. $a + [b - (a - b)]$.

13. $a + b - [(b + d) - (a - b)]$.

14. $m - (n - p) + [3m - 3n - 6m]$.

15. $a - [a - \{a - (-a)\}]$.

16. $x - [3y + \{3z - (z - x) + y\} - 2x]$.

17. $-[m - (m + n) - (m - n) - (-m + n)]$.

18. $12a - \{(a + b) - [b - (a - b)] - a\}$.

19. $2x - \{x - (x - y) - [x - x - y] - y\}$.

20. $12 - 2a - \{-a - [2a - (a - \overline{7 - a})]\}$.

21. $14 - 3a - \{9a - [10a - (11a - 6 - \overline{6a})]\}$.

22. $a - [-\{-(-a)\}]$.

23. $a - 3 - [-\{-(-a + \overline{a + b})\}]$.

24. $x + y - [-(x - y) + \{-x + (x - \overline{x - y})\}]$.

25. $1 - \{-a - (a + 1) - [-a - (a - \overline{a - 1})]\}$.

26. $a - (-\{-[-(-a)]\})$.

27. $1 - (-\{a + (-a + 1)\}) - \{a - \overline{a - 1}\}$.

28. $6m + \{4m - [8n - (2m + 4n) - 22n] - 7n\} + [9m - (3n + 4m) + 14n]$.

29. $1247 - [1722 - \{1722 + (933 - 1247)\}]$.

30. From $a + \{(4 - b) + (a - 4) - \overline{a - 7}\}$ subtract $a - \{(6 - b) + (6a - 6) - (5a - 7)\}$.

31. From the sum of $a + \{a - (b - c)\}$ and $-a + [4a - (5b + c)]$ subtract $a - (b - c)$.

32. Simplify $4a - [6b + (3a - c) - \{5b - c - \overline{a}\}]$ and check the answer by substituting $a = 3$, $b = 2$, $c = 1$ in the question and the answer.

33. Simplify $9a - [-7a + \{5b - (a - b) + \overline{a - b}\}]$ and check the answer by the substitution $a = 1$, $b = 2$.
\section{Exercise 172}
Solve by Cardan's method:

1. $x^3 - 3x - 2 = 0$.

2. $x^3 - 9x + 28 = 0$.

3. $x^3 - 18x - 35 = 0$.

4. $x^3 - 36x - 91 = 0$.

5. $x^3 - 3x + 2 = 0$.

6. $x^3 - 27x - 54 = 0$.

7. $x^3 + 9x + 26 = 0$.

8. $x^3 - 72x - 280 = 0$.

9. $x^3 - 6x + 9 = 0$.

10. $x^3 + 9x - 26 = 0$.

11. $x^3 - 9x - 28 = 0$.

12. $x^3 - 72x + 280 = 0$.

13. $x^3 - 6x^2 - 12x + 112 = 0$.

14. $x^3 + 5x^2 + 8x + 6 = 0$.

\section{Solution of Biquadratics}

594. A biquadratic is an equation of the fourth degree.

595. Descartes' solution of the biquadratic.

Let
\[
x^4 + px^2 + qx + r = 0.
\]
\[(1)\]

Assume
\[
x^4 + px^2 + qx + r = (x^2 + lx + m)(x^2 - lx + n)
\]
\[(2)\]

\[
= x^4 + (-l^2 + m + n)x^2 - l(m-n)x + mn.
\]

Equating the coefficients, we have
\[
-l^2 + m + n = p, \quad -l(m-n) = q, \quad mn = r.
\]
\[(3)\]

Considering that
\[
(m+n)^2 - (m-n)^2 - 4mn = 0,
\]

it is easy to eliminate $m$ and $n$. We obtain
\[
l^6 + 2pl^4 + (p^2 - 4r)l^2 - q^2 = 0.
\]
\[(4)\]

Equation (4) is a cubic in $l^2$, which has always one real positive root. When $l$ is known, the values of $m$ and $n$ are determined from equation (3). The solutions of the two equations

\[
x^2 + lx + m = 0
\]

and

\[
x^2 - lx + n = 0
\]

produce the required roots.
Selecting the cent as the denomination (in order to avoid fractions) we express the statement II in algebraic symbols.

\[
50(11 - x) + 10x = 310.
\]

Simplifying,

\[
550 - 50x + 10x = 310.
\]

Transposing,

\[
-50x + 10x = -550 + 310.
\]

Uniting,

\[
-40x = -240.
\]

Dividing,

\[
x = 6,
\]

the number of dimes.

\[
11 - x = 5,
\]

the number of half dollars.

Check. 6 dimes = 60 cents, 5 half dollars = 250 cents, their sum is \$3.10.

\section{EXERCISE 38}

\begin{enumerate}
\item Two numbers differ by 33, and the greater is four times the smaller. Find the numbers.

\item Find two numbers whose sum is 72 and the greater of which equals five times the smaller.

\item The difference between two numbers is 8, and if 16 be added to the greater, the result will be three times the smaller. Find the numbers.

\item The difference between two numbers is 2, and the difference between their squares is 16. Find the numbers.

\item The sum of two numbers is 47, and their difference is 9. Find the numbers.

\item Divide 20 into two parts, one of which increased by 14 shall be equal to the other increased by 10.

\item One number is 5 less than three times another number. If the second number is subtracted from five times the first number, the result is 25. What are the numbers?

\item Divide 22 into two parts such that one part multiplied by 5 is equal to the other part diminished by 2.

\item Find two consecutive numbers whose sum is equal to 243.
\end{enumerate}
118. The third division will be as follows:

Divisor: $1+x$

Quotient:
\[
1-3x+6x^2-10x^3+15x^4-21x^5+28x^6-36x^7+45x^8-55x^9+66x^{10}-78x^{11}+\&c.
\]

Dividend:
\[
1-2x+3x^2-4x^3+5x^4-6x^5+7x^6-8x^7+9x^8-10x^9+11x^{10}-12x^{11}+\&c.
\]

The division may be exhibited schematically as the continual subtraction produced in the common method of algebraic division:

\[
\begin{aligned}
1-2x+3x^2-4x^3+5x^4-6x^5+7x^6-8x^7+9x^8-10x^9+11x^{10}-12x^{11}+\cdots \\
\qquad -(x- x^2) \\
\qquad\;\; +6x^2-4x^3 \\
\qquad -(10x^3-10x^4) \\
\qquad\;\; +15x^4-6x^5 \\
\qquad -(21x^5-21x^6) \\
\qquad\;\; +28x^6-8x^7 \\
\qquad -(36x^7-36x^8) \\
\qquad\;\; +45x^8-10x^9 \\
\qquad -(55x^9-55x^{10}) \\
\qquad\;\; +66x^{10}-12x^{11} \\
\qquad -(78x^{11}-\cdots)
\end{aligned}
\]

By this division it appears that the fraction $\dfrac{1}{(1+x)^3}$ is equal to the infinite series

\[
1-3x+6x^2-10x^3+15x^4-21x^5+28x^6-36x^7+45x^8-55x^9+66x^{10}-78x^{11}+\&c.
\]

in which, as in the two former quotients, the second, fourth, sixth, eighth, tenth, twelfth, and other following even terms have the sign $-$ prefixed to them, or are to be subtracted from the first term $1$; and the third, fifth, seventh, ninth, eleventh, and other following odd terms have the sign $+$ prefixed to them,
= 0, or till the said series of fractions is terminated, or exhausted. And then it would follow, from what has been shewn above concerning the literal parts of the terms of the said series, that the said series which is equal to $(a+b)^m$, or the $m$th power of the binomial quantity $a+b$, would be

\[
a^m + \frac{m}{1}A a^{m-1}b + \frac{m-1}{2}B a^{m-2}b^2 + \frac{m-2}{3}C a^{m-3}b^3 + \frac{m-3}{4}D a^{m-4}b^4 + \frac{m-4}{5}E a^{m-5}b^5 + \frac{m-5}{6}F a^{m-6}b^6 + \&c.,
\]
continued to $b^m$. Q. E. I.

28. This method of discovering (by a conjecture grounded on some trials in particular examples) that the generating fractions by which the numeral co-efficients of the third, and fourth, and other following terms of the series that is equal to $(a+b)^m$ (or any integral power of the binomial quantity $a+b$), are derived from $m$ (the index of the power to which the said binomial quantity is raised), or from the co-efficient of the second term of the said series (which is always equal to the said index) are $\frac{m-1}{2}$, $\frac{m-2}{3}$, $\frac{m-3}{4}$, $\frac{m-4}{5}$, $\frac{m-5}{6}$, \&c., is suggested by Professor Saunderson, in the second volume of his Algebra, in the chapter on the Binomial Theorem; where the Reader will find a good explanation and illustration of the said celebrated Theorem, by a variety of examples, both in the case of Integral powers, and in the case of Roots and other Fractional powers, and even in the case of Negative powers, and of powers that are both fractional and negative; but no demonstration of it in any case, not even in that of Integral and Affirmative powers.

29. We have now shewn with demonstrative certainty that the literal parts of the terms of the series which is equal to $(a+b)^m$, or the $m$th power of the binomial quantity $a+b$, when
fraction $\frac{N-a^m}{m a^{m-1}}$, (by which substitution the said quadratick equation would be converted into the following simple equation,
\[
a^m + m a^{m-1} z + m \times \frac{m-1}{2} \times a^{m-2} z \times \sqrt{\frac{N-a^m}{m a^{m-1}}} = N,
\]
and then resolving the simple equation thence resulting, to wit, the simple equation
\[
a^m + m a^{m-1} z + m \times \frac{m-1}{2} \times a^{m-2} z \times \frac{N-a^m}{m a^{m-1}} = N,
\]
or
\[
m a^{m-1} z + m \times \frac{m-1}{2} \times a^{m-2} \times \frac{N-a^m}{m a^{m-1}} \times z = N-a^m,
\]
or
\[
z \times \left(m a^{m-1} + m \times \frac{m-1}{2} \times a^{m-2} \times \frac{N-a^m}{m a^{m-1}}\right) = N-a^m,
\]
in the usual way, or by the single operation of Division, which would give us
\[
z=
\frac{N-a^m}{m a^{m-1} + m \times \frac{m-1}{2} \times a^{m-2} \times \sqrt{\frac{N-a^m}{m a^{m-1}}}}
\]
\[
= \frac{N-a^m}{m a^{m-1} + \frac{m-1}{2} a^{m-2} \times \sqrt{\frac{N-a^m}{a^{m-1}}}}
\]
\[
= \frac{2 (N-a^m)}{2 m a^{m-1} + (m-1) a^{m-2} \times \sqrt{\frac{N-a^m}{a^{m-1}}}}
\]
\[
= \frac{2 (N-a^m)}{\frac{2 m a^{2m-2} + (m-1) a^{m-2} \times (N-a^m)}{a^{m-1}}}
\]
The number of species found in SD\#15 was lower than that of SD\#7 (5 and 13, respectively) and showed two species with a higher number of clones per species: \textit{Staphylococcus kloosii/carnosus} (6) and \textit{Bradyrhizobium} sp. (4) (Table 1.0). We suggest that the high number of clones per species, identified in SD\#15, represent lower diversity. In contrast, in SD\#7, we identified a lower number of clones per species, but the representative genera was richer. In conclusion, the number of clones per species was inversely proportional to the identified bacterial diversity. Fifty percent (50\%) of the bacteria identified in SD\#7 are associated with human health issues. These include: \textit{Pseudomonas}, \textit{Rahnella aquatilis}, \textit{Serratia fonticola}, \textit{Sphingomonas}, \textit{Shigella flexneri}, and \textit{Corynebacterium mucifaciens}. In contrast, only one of the species, \textit{Corynebacterium mucifaciens}, identified in SD\#15 is pathogenic to humans. The other four species identified from SD\#15 are part of normal or resident microbiota of animals (\textit{Staphylococcus kloosii}; \textit{Propionibacterium acnes}), soil (uncultured soil bacterium) or plants (\textit{Bradyrhizobium}). Interestingly, by comparing the identified species, it is clear that all the species identified in the light dust event sample (SD\#15) were also present in the heavy dust event sample (SD\#7).

\begin{table}
\centering
\caption{Sahara Dust Sample \#7}
\label{table_1}
\begin{tabular}{llll}
No. of Clones & Matching organism(s) & GenBank Accession number & \% homology \\
1 & Arctic sea ice bacterium ARK9973/Marine sediment bacterium OC-12 / \textit{Bradyrhizobium} sp. & DQ453581 & 99 \\
2 & \textit{Corynebacterium mucifaciens} & DQ453573, DQ453583 & 99 \\
1 & \textit{Modestobacter} sp. Ellin 165 & DQ453589 & 99 \\
1 & \textit{Morganella morganii} & DQ453582 & 97 \\
2 & \textit{Propionibacterium acnes} & DQ453569, DQ453572 & 100 \\
2 & \textit{Pseudomonas} sp. & DQ453576, DQ453579 & 98 \\
1 & \textit{Rahnella aquatilis} isolate k & DQ453577 & 99 \\
1 & \textit{Serratia fonticola} & DQ453570 & 100 \\
1 & \textit{Shigella} sp./\textit{Escherichia} sp & DQ453578 & 99 \\
1 & \textit{Sphingomonas} sp. & DQ453585 & 100 \\
4 & \textit{Staphylococcus kloosii/carnosus} & DQ453571, DQ453584, DQ453586, DQ453588 & 97--99 \\
3 & Uncultured bacterium clone MP10-6-01 & DQ453574, DQ453575, DQ453580 & 99 \\
1 & \textit{Sphingomonas} sp & DQ453587 & 93 \\
\end{tabular}
\end{table}

\begin{table}
\centering
\caption{Sahara Dust Sample \#15}
\label{table_2}
\begin{tabular}{llll}
No. of Clones & Matching organism(s) & GenBank Accession number & \% homology \\
4 & \textit{Bradyrhizobium} sp. & DQ453590, DQ453598, DQ453601, DQ453593 & 99 \\
1 & \textit{Corynebacterium mucifaciens} & DQ453596 & 99 \\
1 & \textit{Propionibacterium acnes} & DQ453597 & 98 \\
6 & \textit{Staphylococcus kloosii/carnosus} & DQ453594, DQ453599, DQ453600, DQ453592, DQ453602, DQ453603 & 99 \\
1 & Uncultured bacterium clone rmv5 & DQ453595 & 99 \\
8 & No Insert & False Positives &  \\
\end{tabular}
\end{table}