Secondly, to multiply an expression consisting of more than one term by a similar expression.

Rule VI.-Write the multiplier under the multiplicand, term under term. Multiply each term of the multiplicand (prefixing proper signs) by the first term of the multiplier, then do the same by the second term of the multiplier, and so on until all the terms of the multiplier have been gone through. Then collect these partial products to form the complete product. (See following examples.) a2 + ab + b2 by a + b a + b

Ex. 3. Multiply

(5) Find the product of a + 26 + 3c and a − 28. (6) Find the product of x2 2ax + a2 and x + a. (7) Multiply a2 2ax + a2 by x3]+ 2ax + a3. (8) Multiply 4ab + 4b2 by 2ab — 3ba. (9) Find the value of (a + b)2. (10) Find the value of (2a + 26 + 2c)3. (11) Find the value of (x + y + 2)2. (12) Find the value of (a + b)3.

Remember, in working the following, that when no sign is expressed between a quantity and a bracket or between two brackets, the sign of multiplication is to be understood. (13) Simplify (a + b − c )(a − b + c). (14) Simplify 6 (3a + 2ỏ)(za — 36). (15) Simplify (a2b + 2abc2)(aba (16) Multiply the sum of 2xy + 2yz and 3xy — 6yz by 2x2y2.

6ab)

The following particular results in Multiplication should be carefully remembered, since their application will save a great deal of labour,

(1) (x + y)2 = x2 + 2xy + y2.

(2) (x
- y)2 = x2
2xy + y2.
(3) (x + y)(x − y) = x2 — y2.

The following examples will show the method of apply ing these results.

Ex. 1. Suppose we wish to find the value of (a + b)a We know from result 1 that

(x + y)2 = x2 + 2xy + y2 and if we write a instead of x, and ỏ instead of y, we get (a + b)2 = a2 + 2a5 + b3.

Ex. 2. Suppose we wish to find the value of (a — 3b)a From result 2 we see that

(x − y)2 = x2 — 2xy + y2 in which, by writing a for , and 36 for y. we get (a — 36j2 · Caž + gi2. = a2 - 2×a× 35 + 962 = a2

2.x

+ 3y

EXERCISE XI.

THE Tweed-up which we purpose taking our imaginary trip this month-forms, during the latter part of its course, the boundary between England and Scotland. It drains the south-eastern counties of the latter country, and falls, after a course of about 100 miles, into the German Ocean.

The first town we see on entering the river is Berwick-on-Tweed, which is an old picturesque town with a pier and lighthouse on the northern bank, still surrounded by fortified walls, and possessing the peculiarity, like Newcastle and some other towns in the kingdom, of forming a county by itself. In the early history of our country its name often occurs. King John stormed the town in 1216, and in the hall of the castle in 1292 Edward I. pronounced judgment in favour of Baliol as King of Scotland. After being retaken by Bruce in 1308, it was surrendered to Edward IV. for good

in 1482. On the opposite side of the river we see the more modern town of Tweedmouth.

Leaving both towns behind us, we proceed up stream, pass the mouth of the Adder on our right, and soon reach that of the Till on the left. The latter stream flows near the battle-field of Flodden, so fatal to the Scotch and their King, James IV., in 1513.

We then arrive at Coldstream, a town which gives its name to the Coldstream Guards, the first regiment bearing that title being raised there by General Monk in 1659. Here also was one of the chief fords or passages across the river, by which, in olden time, incursions were made from the one country into the other.

Still proceeding westward, we enter the county Roxburgh, and reach Kelso, a town at the mouth of the river Teviot,

which, like Berwick, suffered considerably in the early wars between the two countries. In the outskirts are the ruins of a fine old abbey, founded in 1128 by King David I. A few more miles bring us, through some of the prettiest scenery on the river, to Dryburgh Abbey, in which lie the remains of Sir Walter Scott, whose pen has immortalized the whole district through which we are passing.

A small stream called the Leader now flows in on our right, and shortly afterwards we see on the opposite bank the small village of Melrose, with the ruins of another magnificent abbey, commenced in 1326 by Robert Bruce. The river Gala, on which is situated the town of Galashiels, famous for its manufactures of woollen

goods, here joins us on the right, and on the opposite bank stands Abbotsford, the residence of Sir Walter Scott.

We then pass the mouth of the Ettrick, near the head waters of which was the birthplace of the poet, James Hogg, the Ettrick shepherd; and after leaving Innerleithen, a village celebrated for its mineral springs, and mentioned in Scott's works under the name of St. Ronan's Well, we come to the small town of Peebles. Above

this the Tweed trends southward, and, after tracing its course through a picturesque valley, called Tweeddale, shut in by hills on either side, we find the source of the river in Hart Fell, a mountain 2,635 feet high in the extreme south of the county of Peebles.

LESSON XXV.

To keep the bones strong and to mend every place that wears away, we must eat bone-making food, or as we call it, mineral food. Now we get minerals in several kinds of food.

We get it in water. Salt, and lime, and iron, are found in our drinking water, and we eat it with other food in the shape of common salt, and it is found in vegetables in the shape of potash. The quantity of mineral food that is necessary is not very large, so that if our drinking water is pure, and if we are careful to eat sufficient vegetables, we are lmost sure to get enough. If people are unable to get fresh vegetables their blood becomes impure and they have all sorts of skin diseases. When our sailors go long voyages they often suffer because they cannot get enough mineral food by means of vegetables.

Here is a list of our principal mineral foods and the food in which we are likely to find them :Common Salt, found in all animal food, and we also add it to many foods.

Potash, found in seeds that are food, such as peas and beans, and also in fruits and vegetables. Lime, found in water, and in seeds, and fruits. Iron, found in small quantities in nearly all solid foods, also in milk and water.

In every 1,000 lbs. of the following foods there is so many pounds of mineral food only :

nous foods are called Fibrine, Gluten, Albumen, Casein, Legumin, and Gelatine.'

Now these are the foods that make flesh, and they are called by different names b cause they are found in different places. though some of them are very much alike. For instance, casein is found in cheese, and legumin is found in peas and beans. This last is often called 'vegetable casein,'and is really more easily diges:ea than the first.

Albumen is found almost pure in the white of an egg, in the juice of most vegetables, in the blood of animals, and in most seeds.

Fibrine is also found in vegetables and in all ñen foods; also in wheat grain, where it is called gluten.

Gelatine is found in flesh, fibre, and tissue.

Casein in the curd of milk, of which we mal.e cheese.

Legumin in peas and beans.

LESSON XXVII.

OUR warmth-giving foods are of three kinds and have three different names, as you have learned. 1. The Farinaceous or floury starchy foods. All foods which contain starch or flour come under this head-flour and meal, rice, coraflour, sago, and arrowroot, as well as many others.

2. The Oleaginous foods. These are those that contain fat in any form. Butter, milk, the fat of meat, nuts in which there is much oil. and many others.

3. The Saccharine or sugary foods. Sugar is fou:Ja in almost everything. Some vegetables contain large quantities of it, so much, indeed, that we make sugar from them, as in the case of the beetroot. All fruits, too, contain sugar, and we have besides the sugar-can. Here is something which a very learned man tells us about sugar :

'Grape sugar is found in all vegetables. It can be made from starch, as well as from cane sugar, and by the action of strong acids we can procure it from paper, old rags, and some woods.'

A gentleman once said, that 'if all other sources failed, enough sugar could be procured for the whole world from sawdust only.

STANDARD III.

D. 1. Twenty million twenty thou sand and twenty ÷ 89,607.

2. Take 7395 pence from £60.
3. From 870 cr. take 807 fl.

2.

E. 1. How many threepences are there in 85,763 guineas? Add together 7 each of all our gold and silver coins. How much are £675, 3827 cr., 8017 fourpences, and 7164 threepences?

3.

and twelve guineas and fourpence.

2. Divide 367,801,490 by 56,080. . From £500 take 175. 10d.

A. 1. £973 15s. 8d. x 35.

2. £789 16s. 10d. x 96. 3. £7689 17s. 6d. ÷ 17. B. . Multiply ninety-seven pounds thirteen shillings and ninepence farthing by 132. 2. From £76,809 17s. 8d. take £9109 18s. 94d., and divide the remainder by 807. 3. How many seconds are there from July 29th at 7.25 p.m. to the end of the year? C. 1. £768,241 18s. 3 d.÷9083. 2. £83 17s. 9 d. x 365; bring the product to farthings.

3. Bring eight million twelve thousand and seventeen ounces to tons, cwts., etc.

remainder of 685; what is the true remainder ? 2. 916,823 farth.-4127 flor. 3. 91,782,426÷(4689 × 15).

E. 1. 51 mls. 5 fur. 36 pls. at £17 18s. 8d. a mile.

2. A bill-3 yds. at 7s. 7d., 91 at 8s. 3d., 121 at 4 d., 38 at 2s. 4 d., and 865 at 24d. 3. If 2 lbs. of tea cost 6s. 2 d., how many lbs. could be bought for £19 145. 7.? F. 1. A bill-3 cwt. at 4d. a lb., 7 tons at 3. 6d. a cwt., 18 lbs. at 10дd. a lb., 41 cwt. at 7 12s. 6d. a ton, and 126 lbs. at 14s. 1od. a cwt.

2.

ac.

3. If 5 rabbits cost 8s. 61d., what would 131 cost?

3.

What would 83,750 articles cost at £3 2s. 6d. a 1000? 96 tons 16 cwt. I qr. 18 lbs. at £5 16s. 8d. a ton.

ADVANCED EXAMINATIɔn.

1. A dealer bought 60,000 eggs at 9d. a score, and sold them at 7d. a dozen; how much did he gain or lose?

2. Out of £50 a man bought a horse for 42 guineas, and then spent the remainder, except 12s. 63., in a saddle and bridle, the saddle cost 3 times as much as the bridle; find the cost each of the saddle and the bridle.

3 A boy earns 4s. 9d. a week, and his father earns 4s. 6d. for every shilling that he earns; how much do they both together earn in a week?

ADVANCED EXAMINATION.

I. A draper bought 600 yards of silk for £87 12s. 6d. He sold half of it at 4s. 6d. a yaid, 140 yards at 3s. 9d., 36 yards at 2s. 9d., and the remainder at 25. 3d.; how much did he gain?

2. A field of 20 acres was mown by 6 men in 5 days of 8 hours each, and they were paid at the rate of 7s. 6d. per acre; how much did each man earn per hour? 3. A traveller, who left home August 4th at 7.45 a.m., returned after an interval of 71,609 minutes; give the exact date of his return home.

ADVANCED EXAMINATION.

1. Divide £52 10s. among 18 men, 12 boys, and 60 girls, so that a man shall receive six times as much as a boy, and a girl one and a half times as much as a boy. What will each girl get? 2. Make out neatly this bill-18 yds. of silk at 6s. 9d., 163 yds. of linen at Is. 44d., 8 yds. of cloth at IIS. 9d., 34 yds. of velvet at 16s. 3d., and 28 yds. of carpet at 35. 9d.

3. If 10 men or 15 women can do a piece of work in 24 days, in what time could 8 men and 6 women do it?