cients, it is obvious that we have

=

− (y + px)2}

= 2ƒ {r2q (1 + p2) − p (2p + qx) (y + px)
+ (2p2— qy) (y + px)}

Squaring both sides, and substituting for p, q, f, their values and simplifying, we have finally

I +

(

dy

+x)']

+ x2 dx

dx This is a differential equation of the second order and twelfth degree, and consequently will in general involve twelve ambiguities and two undetermined constants.

The ambiguities, however, may be disposed of, because the limits of integration are in all cases x and o.

It is humiliating, however, to have to confess that here the powers of mathematical analysis can carry us no farther. There is no process yet known by which such an equation as we have obtained can generally be solved. Throughout the whole range of this subject of differential equations, indeed, we are on the boundaries of our knowledge. The merest novice could puzzle the greatest expert. In what way the exploration is to be com menced, or whether it has any chance of ultimately, and in reasonable time, succeeding, is more than can be said. Even algebraical equations still remain a puzzle; quadratics, cubics, and quartics have long been mastered, but to find an algebraical solution of the general equation of the fifth and higher orders has, as yet, baffled the acutest mathematicians. The case is still worse with differential equations. No one has yet mastered the analogues to quadratics even, except in very special and limited cases, much less those of higher orders.

Thus, then, it is completely evident that we must admit ourselves beaten by the horse; nevertheless, as the result of geometry or obvious analysis, we may state the following facts :

(1) If the relative velocities of horse and man be such that the man catch the horse before the first round, the curve will be as in the fig., where P1, B1; P2, B2; P3, B3; P4, B1 represent successive simultaneous positions of man and horse respectively, P1 and B, being coincident.

(2) If the velocities of man and horse be such that the man do not catch the horse before it have run once round, he will never catch it; in general his path will always be approaching indefinitely nearer to an asymptotic circle of radius er.

Of course when eo, i.e., velocity of horse is infinite, the man never leaves the centre.

The whole subject of curves of pursuit, as they are called, is at present in a very imperfect state, owing chiefly to the imperfection of our tools. If we suppose the circle to be of infinite radius, and the man to have run so far, having started from the centre which is at infinity, as to be now a finite distance from the horse, the problem reduces itself to one that is treated in Tait and Steele's Dynamics of a Particle. The question is there put under the form of the curve which a dog takes in following his master. It is shown that if e be the ratio of velocities of man and dog, the Cartesian equation is

the latter curve having yo as an asymptote.

One is almost tempted to doubt whether the dog is fully conscious of the niceties of his path.

This particular case is the only one which has yet been satisfactorily investigated, and of course it fails to give the numerical answer to your question.

16. PAN, Hackney.-(1) No. (2) Whenever you have a doubt about the interpretation of directions in an examination, take them in the widest, and therefore safest manner. In the case you name, decidedly state the rules of concord.

17. TUTBURY, notwithstanding Mason, is perplexed about the phrases-

(1) An Accusative. (2) A Genitive.

(3) A Dative.

There would certainly be five other theories if there happened to be five cases on to which to tack them. The third we may dismiss at once. If the second be the right one, we may perhaps say that the root-idea, arising in an obvious manner, is partition. If the first, that the root-idea is extent or measurement. Both these notions are represented in Latin, similarly, by Genitive and Accusative. If Tutbury' is still not satisfied, and wishes to know further why the Genitive expresses partition, the Accusative extent, we must frankly say we do not know. The ideas of partition and extent existed before the aborigines, who, finding them, decreed special inflections to represent them, which we in turn, discovering, christen Genitive and Accusative. 18. A SUBSCRIBER, Manchester.-Out of a bag of silver I take 50s. more than 5 of whole sum which it contained; then 30s. more than 2 of what remained; and then 20s. more than 25 of what then remained; after this 10s. remained. What did the bag contain at first? Turn the sum upside down.

20. W. J. WHITE, Reading.-Divide a straight line into two parts, such that the square on the one part shall be equal to twice the square on the other part. Let AB be the given line. Draw AD perpendicular to AB, and bisect the angle DAB by AE. Bisect the angle EAB by AF. Make ABFI equal to half a right angle, taking F1 on the same side of AB as F was, and let AF and BF intersect in F. Make [AFIC equal to the angle FAC, and let FC meet AB in C. Then C shall divide the line as required for CB2 = CF2 + FB2 = 2CF2 = 2CA2.

21. L. N. M., Corrie.-If the increase in the number of male and female criminals be 1.8 per cent., while the decrease in the number of males alone is 4'6 per cent., and the increase in the number of females is 98, compare the number of male and female criminals. f standing for the number of female criminals, m for the number of males,

22. J. F. M., Bryn Amman.-If a snail creep 2 feet 7 inches up a pole during 12 hours in the night, and slip down 16 inches during 12 hours in the day, how many hours will he be getting to the top of a pole 35 feet high? In 26 days he will have climbed 26 (2 1}}} feet 26 × feet 32 feet,... there are 2 feet 6 inches left. These he will do in × 12 hours. since he does 31 inches in 12 hours, ... altogether he takes 6351 hours. Your difficulty has been the same as that of the sophist to whom the question was first proposed, reasoning thus: Net result of his climbing is 1 feet per diem, he has 35 feet to do, ... he takes 28 days.

23. C. P. EDWARDS, Glasgow.-Use the following books :Latin Prose through English Idiom, by Dr. E. A. Abbott (Seeley), 2s. 6d. ; Hamblin Smith's Latin Composition (Rivington), 3s. 6d. ; Arnold's Latin Prose Composition, Part I, 6s. 6d.; Millington's Selections for Latin Prose (Longman), 8s. 6d. If these are successfully worked off, you may proceed to Wilkins' Selections for Latin Prose. Get the books in the order named ; but Arnold's is the best if you only wish one.

24. JAMES CARPENTER.-What are the factors of

=

Thus + 643 x2 + (423 (x + 4y) (x + 4wy) (x + 4w2y) 729x3+51233=(9x)3 + (6y)3x = (9x+6y) (9x+6wy) (9x+6w2y). If real factors only are wanted, they are respectively

(x+4y) (x2 - 4xy + 1612) and (9x+6y) (81x2 - 54xy +36μ3). 25. J. S., Dumfries. It would seem to us that you already know amply enough to get you creditably through the examination to which you refer. If you desire a further enlargement of your views on the subjects, the following are rather more advanced books:-Todhunter's Euclid (Macmillan), 3s. 6d. Read to the end of Book VI., so as to enable you to grasp the Trigonometry more perfectly, and work as many of his 600 exercises as you can. Todhunter's Algebra (Macmillan), 7s. 6d. You can omit most of the chapters which do not bear on your work. Todhunter's Trigonometry (Macmillan), 5s. Same remark applies to this. Sonnenschein and Nesbitt's Arithmetic (Sonnenschein), 5s. 6d. This will be of particular help if you are at all shaky in elementary notions of number, quantity, etc. We have not space to give you any general directions for work, but must refer you to the article on How to Write an Examination Paper,' in Number 1, PRACTICAL TEACHER.

26. J. GREEN, Stockport.-The average of ten results was 17; that of the first three was 161, and of the next four 161; the eighth was 3 less than the ninth, and 4 less than the tenth. What was the last result?

Let x be the last result, x-4 is the eighth result,
I is the ninth;

also sum of first three

..sum of last three=x+(x-4)+(x − 1) = 8x − 5, three times their average=3× 1614=83, next four four times their average = 4 × 161=66, and sum of all the ten ten times their average = 175; ..175=66+48 +3x-5=3x+109; 3x=651, x=21.

27. R. GIBBONS, Stockport.-The sum of £1001 was laid out in the 3 per cents. at 89, and a whole year's dividend having been received upon it, it was sold out, the whole increase of capital being 72 guineas. Find at what price it was sold out. £89 produces an income of £3,

50 men

I c. foot in

162 x 20 III X4×9 18 5 6

hours.

8

,, 300 × 8 × 24 c. feet in 162 × 2 × 8 × 8×4 hours.

=933 days=93 days 4 hours nearly, inasmuch as each day consists of 9 hours nearly. (2) Suppose the labour to increase uniformly as the depth increases, instead of at one moment. In this case it may be proved in the same way as s=ft2 is established in Dynamics that the total increase of work is just double what it was, so that the labour is equivalent to that due to a depth of (1+) × 18 feet = × 18 feet; . the time employed is of what it was, i.e. 116 days 7 hours nearly.

30. NIHIL, Whitchurch.-Square 19 19s. 113d. in a shorter way than by Vulgar Fractions or Practice. Though we must confess ignorance as to what a square £ sterling is, it is obvious what our correspondent means.

28. J. H., Cranbrook.-(1) We do not advise you to try to learn Mensuration without some knowledge, however slight, of Algebra. The very problems that occur in practice are the ones that need a little Algebra; take, for example, the one that we answered last month about a hole bored through a sphere. Before you commence Mensuration we strongly advise you to know (i.) the rudiments of Algebra; (ii.) Euclid, Books I.-VI.; then you may enter on the study of Todhunter's Mensuration (Macmillan), price 3s. 6d. Even without the knowledge we name, we think Todhunter would meet your purpose. (2) Hamblin Smith's Algebra, price 35., key 9s.., would suit you best. The publisher is Rivington.

29. C. WHATMOUGH, Stockport.-If 20 men can dig a trench 37 yards long, 4 feet wide, and 9 feet deep, in 18 days of 9 hours each, how many days of 10 hours each will 50 men require to dig a trench 100 yards long, 8 feet broad, and 18 feet deep, the labour increasing by on the average for every 9 feet of depth? This sum is worded in such an indefinite way that we cannot certainly determine what it means. We work it under the two different interpretations. (1) The ordinary arithmetical sensethe work increasing by at the end of 9 feet; hence, as the depth is 18 feet, the total work done is that due to a depth of (1+) × 18=24 feet.

Engagements for July.

July 1. Council of Teachers' Orphanage and Orphan Fund. 2. Executive Meeting and Organisation Committee of N.U.E.T.

4. Royal Asiatic Society.

Education Society, 'Rewards and Punishments.' Discussion.

5. General Board of Management of Teachers' Provident Society.

6. Parliamentary and Law Committee of N.U.E.T. Entomological Society.

7. Zoological Society, 'The Limbs of Birds.' Prof. W. K. Parker, F.R.S.

11. Education Society, 'Object Lesson.' T. M. Williams, B. A.

13. Finance Committee of N.U.E.T.

14. Zoological Society, 'Birds Ancient and Modern.' W. A. Forbes, Esq.

15. Executive Meeting of N. U.E.T.

16. Organisation Committee of N.U.E.T.

18. Central Committee of Teachers' Benevolent Fund.
20. Parliamentary and Law Committee of N.U.E.T.
21. Royal Historical Society. Papers by the Rev. E. King
and Dr. Zerffi.

Zoological Society, 'Zoological Gardens.' P. L. Sclater,
Esq., F.R.S.

25. Finance Committee, N.U.E.T.

28. Zoological Society, 'Chameleons.' Prof. Mivart, F.R.S.

'No one can read these pages without being convinced that a vast improvement ought to be effected in many of the methods of performing the most ordinary operations of Algebra. More tha this, no beginner can work through the examples given in the book, according to the plans laid down in it, without bringing to all future work a facility of handling and transforming Algebraical expressions formerly quite wanting.-Sheerness Guardian.

'You may freely use my name in recommending the book. Send me 50 copies.-J. B. COLGROVE, M.A., Head Master, Grammar School, Loughboro'.

'I think it a pity that all teachers of Algebra do not know there is such a capital book to be had at a very moderate price.'-W. J. MANTLE, F. R. G.S., Head Master, Diocesan School, Lincoln. Other remarks by practical teachers :-'It is all it professes to be.' 'Valuable and not costly.' That rara avis in this stereotyped age-something both new and good.' 'A real boon to all teachers of Algebra.' 'Your admirable book.'

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