difference 5, and the number of terms 12, what is the last term? Ans. za (n − 1) d = 60 – (12 - 1) × 5 = 5.

(3.) If the first term of an ascending series be 9, and the common difference 4, what will the 5th term be? Ans. z = a + (m − 1) × d = 9 + (5 − 1) × 4 = 25.

There is one other inquiry to be made concerning a series in arithmetical progression. It is often necessary to find the sum of all the terms. This is called the summation of the series. The most obvious mode of obtaining the amount of the terms is to add them together. But the nature of progression will furnish us with a more expeditious method.

Let us take, for instance, the series and also the same inverted,

3, 5, 7, 9, 11. 11, 9, 7, 5, 3. 14, 14, 14, 14, 14. a+d, a +2d, a + 3d, a + 4d, and the same inverted, a + 4d, a + 3d, a + 2d, a + d, `a

The sums of the terms will be, Take also the series a

A variety of other formulæ may be deduced from the equations already given, the investigation of which will afford the student a pleasing and profitable exercise.

By the third formula, for example, may be found any number of arithmetical means between two given numbers. For the whole number of terms consists of the two extremes and all the intermediate terms. If, then, m = number of means, m + 2 =n, the whole number of terms. Substituting m + 2 for n in the third equation, we haveThe common difference, d= given. EXAMPLE.-Find 6 arithmetical means between 1 and 43.

m+1'

in which a, z, and m are

=

2 -a 43-1 Here n = 8; a = 1; = 43; d= = 6, comN- - 1 8-1 mon difference; .. the series is 1, 7, 13, 19, 25, 31, 37, and 43. It is obvious, from the mode in which we obtained an expression for the sum of an arithmetical series, that the sum of the extremes is equal to the sum of any other two terms equally distant from the extremes. Thus, in the series 3, 5, 7, 9, 11, the sum of the first and last terms, of the first but one and last but one, etc., is the same in each case, viz., 14. The same is true of every series.

EXERCISE 68.

1. If the first term of an increasing arithmetical series is 3, the common difference 2, and the number of terms 20, what is the sum of

2. If 100 stones are placed in a straight line, at the distance of a yard from each other, how far must a person travel to bring them one by one to a box placed at the distance of a yard from the first stone? 3. What is the sum of 150 terms of the series

we shall have this equation, 8 =

a + z x n, con2

The two formulæ, z = a + (n-1)d, and s = tain five different quantities; viz., a, the first term; d, the common difference; n, the number of terms; %, the last term ; and s, the sum of all the terms.

From these two formulæ others may be deduced, by which, if any three of the five quantities are given, the remaining two may easily be found. The most useful of these formulæ are the following:

:

4. If the sum of an arithmetical series is 1455, the least term 5, and

the number of terms 30, what is the common difference?

5. If the sum of an arithmetical series is 567, the first term 7, and the common difference 2, what is the number of terms? 6. What is the sum of 32 terms of the series

1, 11, 2, 2, 3, etc. ?

7. A gentleman bought 47 books, and gave 10 shillings for the first, 30 shillings for the second, 50 shillings for the third, etc. What did he give for the whole ?

8. A person put into a charity-box a shilling the first day of the year, two shillings the second day, three shillings the third day, etc.,

to the end of the year. What was the whole sum for 365 days? 9. How many strokes does a common clock strike in 24 hours? 10. The clocks of Venice go on to 24 o'clock; how many strokes do they strike in a day?

11. Required the sum of the odd numbers 1, 3, 5, 7, 9, etc., continued to 100 terms; and also to n terms.

12. Required the 365th term of the series of even numbers 2, 4, 6, 8, 10, 12, etc.; and also the nth term.

13. The first term of a series is 4, the common difference 3, and the number of terms 100. What is the last term, and also the nth term? 14. A man puts £1 out to interest at 6 per cent.; what will be the amount in 40 years at simple interest ?

15. The extremes of an arithmetical series are 2 and 29, and the What is the common difference?

3. The common difference, d= Transposing and dividing, 4. The number of terms, n = By the second formula, 5. The sum of the terms, s= Or, by substituting for its value, 2a + (n-1) d Xn; in which a, n, and d are given. 2 Reducing the preceding equation, 28-dn2 + dn

8 =

6. The first term, a =

2

2n

7. The common difference, d given.

8. The number of terms, n = and s being given.

a + z 2

=

reducing the equations. Thus :

;s, d, and n being given.

Let r the second of the four numbers,

And y = their common difference.

The series will be x-y, x, x + y and x + 2y.

By the conditions, (x − y) + x + (x + y) + (x + 2y) = 56.
And (xy)2+ x2 + (x + y)2 +(x+2y)2= 864.
That is, 4x + 2y = 56.

And 4+4ry + 6y2 = 864.

Reducing these equations, we have x = 12, and y = 4. The numbers required, therefore, are 8, 12, 16 and 20. EXAMPLE. A certain number consists of three digits, which are in arithmetical progression, and the number divided by the

sum of its digits is equal to 26; but if 198 be added to it, the digits will be inverted. What is the number?

Let the digits be equal to x- -y, x, and x+y, respectively. Then the number = 100 (xy) + 10x + (x + y) = 111x - 99y, etc. This example will give the result=234.

EXERCISE 69.

1. The sum of three numbers in arithmetical progression is 9, and 2. The sum of three numbers in arithmetical progression is 15, and

the sum of their cubes is 153. What are the numbers ?

the sum of the squares of the two extremes is 58. What are the numbers ?

3. The sum of the squares of the extremes of four numbers in arithmetical progression is 200, and the sum of the squares of the means is 136. What are the numbers ?

4. There are four numbers in arithmetical progression; the sum of the squares of the first two is 34, and the sum of the squares of the last two 130. What are the numbers ?

5. There are four numbers in arithmetical progression whose sum is 28, and their continued product is 585. What are the numbers?

THE arteries are distributed to all parts of the body; the only portions which are destitute of them are the hair, the nails, the outer covering of the skin, and the cartilages. They divide and subdivide, the branches freely communicating with each other, till they become diminished to a very small size, and at length they terminate in a very delicate network of vessels, which, from their great minuteness, are termed capillaries (or hairs). The great artery of the body, called the aorta, starts from the right ventricle of the heart, and first ascends, and makes a kind of arch; it then descends, passing downwards through the thorax and abdomen, at the lower part of which it divides into two. From the arch of the aorta are given off large branches --the innominate artery, which divides into the right common carotid, and the right subclavian. The common carotid ascends the side of the neck, and divides into the external and internal carotids. From the first of these numerous branches arise, which are distributed to the external parts of the neck, the head, and the face; the internal carotid passes up into the skull, and is the principal channel for the blood going to the brain. The subclavian gives off a large branch, the vertebral, which enters the skull through the occipital foramen, and completes the blood-supply for the cerebral organs; it then gives off branches to the shoulder and external parts of the chest, and terminates in a large trunk called the axillary, which afterwards, taking the name of brachial, passes down the arm to the bend of the elbow, where it divides into the radial and ulnar arteries, which supply the fore-arm and hand. The left common carotid and subclavian arise directly from the aorta, without the intervention of an innominate artery. From the descending trunk of the aorta branches spring which supply all the viscera and the muscular walls of the thorax and abdomen, and eventually the aorta splits into two halves, called the right and left common iliacs; these each again divide into external and internal iliacs, the latter of which is distributed to the pelvic organs, whilst the former, taking the name of the femoral, at its exit from the abdomen, passes down the front of the thigh, giving off branches to the muscles in that neighbourhood; twothirds down it pierces the muscles, and appears at the back just above the bend of the knee, there called the popliteal. Soon after it enters the leg, it divides into two, an anterior and posterior tibial, which supply the leg and foot. This is the general arterial circulation of the body; but in addition to this must be mentioned the pulmonary artery, which springs from the left ventricle, and immediately divides into a right and left pulmonary artery; these convey the blood to the corresponding lungs, in the substance of which they break up into a dense network

of capillaries, which will be more particularly described when we come to speak of the structure of the lungs.

The capillaries, or intermediate vessels in which the finest branches of the arteries terminate, are extremely minute, their average diameter being about of an inch: they vary somewhat in size; those of the brain and the intestines are the smallest. These vessels form a dense network all through the body, their number and the closeness of the network being proportionate to the activity of the tissue they have to supply with blood; the walls of the capillaries are composed of a fine transparent membrane, containing cells interspersed at intervals, and offering little obstruction to the process of absorption. There is no definite line to mark where vessels cease to be arteries and become capillaries, or where the veins commence ; but the intermediate vessels have this peculiarity, that when once they have attained a certain degree of minuteness, they retain it, and do not continue to diminish, and the meshes of the capillary network are more even and uniform than those formed by the smaller branches of the arteries or the commencing radicles of the veins. The veins take their rise from the capillary network, first as very small vessels, and gradually join together, forming larger and larger trunks, till they are all eventually merged in two, which have been already mentioned, the superior and inferior cavæ. The veins are larger and more numerous than the arteries, and convey back to the heart the blood which has exhausted its nutritive properties. In structure their walls resemble those of the arteries, but have very little elastic tissue in them; in shape they are not so completely cylindrical as the arteries, and when empty their walls collapse: they also have another important point of difference from the artery, in that there are valves placed in all the larger veins that are subject to much pressure. These valves, which are semi-lunar in shape, and generally occur in pairs, are so arranged as to allow the blood to pass onwards towards the heart, but prevent any backward movement of the current. Veins may be divided into superficial, deep, and sinuses. The superficial lie immediately beneath the skin, and communicate with the deep ones. The deep veins accompany the arteries, and are usually inclosed in the same fibrous sheath; to the larger arteries, such as the femoral or the subclavian, there is but one vein to each artery; but in the smaller ones, as the radial or ulnar, there are a pair, one lying on each side of the artery. In the brain, and some other parts of the body, the arteries and veins take different courses, and do not accompany each other. The venous sinuses only exist in the interior of the skull; they are large channels, formed between the layers of the dura mater, which collect the venous blood from the substance of the brain and discharge it into the internal jugular veins.

Having now examined the blood, and the apparatus by which it is circulated, we pass on to consider the act of circulation itself, and we may take as a starting-point the left ventricle of the heart. When this chamber is filled with blood, it contracts and forces the blood into the aorta; this conveys it, by means of its many branches, to all parts of the body. When the blood has reached the extreme divisions of the arterial system, it leaves them and enters the capillary network; from thence it makes its way to the ultimate radicles of the veins, which carry it forward and empty it into the superior or inferior cava; these at their termination empty it into the right auricle of the heart; the auricle, when it is filled, contracts and drives the blood into the right ventricle, which in its turn pumps it into the pulmonary artery; this vessel, dividing into two, conveys it to the lungs ; here, whilst passing through the capillary network, it is exposed to the action of the air; leaving the lung, it is conveyed by the pulmonary veins and discharged into the left auricle, which contracting, drives it once more into the left ventricle, to commence again the same unceasing round.

In addition to the general circulation of the body, there is a minor one of the liver, called the portal circulation. This has been before alluded to in the article on Digestion. The veins which collect the blood from the viscera of digestion join together to form a large trunk, called the portal vein; this vessel enters the substance of the liver at its under surface, and divides like an artery into a carlary network, thus bringing the blood it conveys into intimate relation with the secreting cells of the liver. This network terminates in a number of moderate-sized veins called the hepatic veins, which unite into three large branches, and finally empty themselves into the inferior vena cava.

We must now consider the part which each constituent of the circulatory apparatus plays in the performance of this function; and first in importance is, of course, the heart. In order to understand the way in which the heart fulfils its duties, we must constantly bear in mind that the heart is a muscular organ, split up into four distinct chambers, and richly supplied with nerve power. The action of the heart is made up of two sets of motions, the dilatation and contraction alternately of the auricles and ventricles: the auricles contract together in alternation with the contraction of the ventricles, which is also simultaneous; the dilatations follow the same rule, and the contraction of the auricles takes place at the same moment that the ventricles are dilating, and vice versa. The interval between the two sets of movements is, of course, very short, but is easily made out when the heart is acting quietly. During the contraction of the ventricles the apex of the heart is drawn upwards and tilted forwards, striking the parieties of the chest, thus giving that sensation which is described as the beat of the heart, and which in a healthy state is usually felt between the fifth and sixth ribs. When the action of the heart is examined by the ear, two sounds are heard. The first is dull and prolonged; its commencement coincides with the impulse of the heart, and just precedes the pulse at the wrist; the second is a shorter, sharper sound, which follows the pulse. The cause of the first of these sounds is still very uncertain; it coincides in point of time with the contraction of the ventricles, and is probably partly caused by the noise or bruit produced by the contraction of muscular fibre. The second sound is held to be occasioned by the sudden tightening of the valves when they are pressed across the orifices of the aorta and pulmonary artery. The contraction of the auricles is a much more rapid and less complete process than that of the ventricles; the auricles are probably never completely emptied; but the ventricles contract so firmly, that in some cases, where the heart has been examined after death, their cavities have been found completely obliterated, only a slight fissure marking their existence. The heart, then, by its contraction propels the blood, and the amount of force thus generated is sufficient to carry the blood through the complete circle. This force has been estimated to be equal to a pressure of six pounds to the square inch; and taking the area of the heart at ten inches, this would give a propelling force of sixty pounds. The left ventricle, as would be supposed, from the much greater thickness of its walls, contracts with a force nearly double that of the right.

The time required for the blood to traverse the circulatory system is very brief, the average probably being about a minute, though in some experiments made by injecting substances into the vein of an animal, the circuit was completed in a much shorter time. By the contraction of the several cavities of the heart the blood is forced along its proper course; but as these contractions are intermittent, and alternate with each other, there would be a constant reflux of the blood into the cavity it had left, but for the interposition of those valves which have been before described. Those which shut off the communication between the auricles and ventricles are the most important, and may be taken as types of the rest. In speaking of them, it was said that their under-surfaces are connected by strings, the chorda tendina, with the summits of the projecting masses of muscular fibre, the columnæ carneæ. When the ventricles contract upon their contents, the blood presses up the flaps of the valves, and so mechanically closes the auricular opening; but this is not all, for just in proportion to the amount of pressure made by the blood upon the under-surface of the valves is the action excited in these little muscular columns, which at once contract and draw the valves tighter and tighter, and close more perfectly the opening, and so prevent the reflux of the blood; this is, therefore, a most perfect floodgate-not simply a mechanical contrivance, but a vital organ, developed just sufficiently to perform the necessary work.

LESSONS IN FRENCH.-LXXII.

§ 63. THE PARTICIPLE.

(1.) The participle is so called because it participates of the nature both of the verb and of the adjective. It partakes of the nature of the verb in having its signification and regimen, and of the nature of the adjective in relating, like the latter, to ns and pronouns.

1. Of Mannor.-Doucement, softly; sagement, wisely; etc. 2. Of Order.-Premièrement, first; d'abord, at first; ensuite, ofter wards; etc.

3. Of Place.-Ici, here; où, where; là, there; ailleurs, elsewhere; etc. 4. Of Time.-Hier, yesterday; aujourd'hui, to-day; demain, tomorrow; etc.

5. Of Quantity.-Peu, little; trop, too much; tant, so much; etc. 6. Of Comparison.-Plus, more; moins, less; très, very.

7. Of Affirmation, Negation, and Doubt.- Oui, yes; certes, certainly; non, no; nullement, by no means; peut-être, perhaps; ne, pas, point, not; etc.

(3.) A few adjectives are sometimes used adverbially. They are then invariable :