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ANSWERS TO THE MATHEMATICAL QUESTIONS.

QUES. 84.- Proposed by J. H. Four labourers, A, B, C, and D, are engaged to dig a field containing 2 acres, 3 roods, 4 perches, 9 sq. yards. A digs 200 yards the first day, 205 the second, 210 the third, and so on; B digs 220 yards the first day, 216 the second, 212 the third, &c. ; C digs 180 yards the first day, 186 the second, 192 the third, &c.; and D digs 225 yards the first day, 220 the second, 215 the third, &c. How many days will the work occupy them? Answered by Mr. Henry Hill, Mr. J. W. Spedding, Mr. Dungate,

and Mr. G. Barnacle.
No. yds. dug in the first day = 825.

second day = 827.

third day 829. Now these numbers are in arithmetical progression, the common difference being 2. Let x = the number of days, then we have by the general formula for the sun of an arithmetical series

{2
2 * 825 + (x − 1)2}

– 1)2} x = 13440,
... ma + 824 x = 13440,

16, the number of days required.

Ques. 85.Proposed by the Editor. Give a geometrical construction of the problem contained in question 81. Answered by Mr. Righton, Prismoid, Mr. Salter, Mr. O'Clazey,

Mr. Royds, Nero, and Mr. Stanwick. Let A B C be the triangle. Upon the base A B describe a semicircle ; divide A B into three equal parts in the points a and b; and from a and b draw the perpendiculars a c and 6 d, meeting the semicircle in the points c and d. With the radius A c cut A B in D, and with the radius A d cut A B in E: then D and E are the points required. Draw D P and E H parailel to B C, meeting the side A C in the points P and H.

Demonstration. A B. A a = A D2, (Euc. VI., 8);
A A D P: A A B C ::A D2 : A Boy (Euc. VI., 17)

:: A B.A a : A B2
:: Aa : AB

::1 : 3,

.. 3 A ADP = A A B C. In like manner 3 A AEH 2 A A BC; and so on.

Ques. 86.- Proposed by John C., Ecclesall School, Sheffield.
Find the value of x in the following equation :

1

8x + gx= 41. Answered by Mr. Herbert, Mr. Salter, and the Proposer. Multiplying by 84, and transposing,

17 827

.8* 1, 4

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Prismoid, Tewkesbury, ans. 84, 85, 86; J. Herbert, Little Woolton, ans. 84, 86 ; J.O'Clazey, Durham, ans. 84, 85, 86 ; Emma Royds, Hollingworth Free School, ans. 84, 85, 86; H. Hill, Chester, ans. 84, 86; W. Righton, Ripley, ans. 84, 85, 86; T. Dungate, Ightham, ans. 84; J. W. Spedding, Bradford, ans. 84, 86; W. Leonard Brown, Ightham, ans. 84; J. Webb, Ightham, ans. 84 ; J. Brown, Barton, ans. 84, 86 ; J. Royds, Belfield, ans. 84, 85, 86; W. H. Levy, Shelbourne, ans. 84, 85, 86 ; J. Stanwick, Gillingham, ans. 84, 85, 86; J. C. Ecclesall, Sheffield, ans. 84, 86; G. Barnacle, Empingham, ans. 84, 86 ; J. Fox, Burton, ans. 84, 86; T. Sothern, Burton-wood, ans. 84, 85, 86; W. Crow, Lockwood, ans. 84, 86; Thomas Brown, ans. 84, 86 ; J. Salter, Durham, ans. 84, 85, 86; A. Ford, Norham, ans. 84, 85; C. P., ans. 84, 85, 86; W. Wakely, Marlborough, ans. 86; Kerslake, ans. 84, 86; Nero, Berwick, ans. 84, 85, 86; G. Morris, Gosport, ans. 84, 86; A. M., ans. 85, 86; J. Sheppard, Gillingham, ans. 84, 85, 86; Quere, ans. 84, 85, 86; Sam. Dyer, Wanstead, ans. 84, 86.

NEW QUESTIONS,
TO BE ANSWERED IN OUR NUMBER FOR FEBRUARY, 1851.

Ques. 87.- Proposed by Mr. O'Clazey, Durham. A man and a boy are engaged to dig a trench. The man could finish it in 12 days, and the boy in 20 ;-supposing them to begin at the same time, one at each end of the trench, when should they exchange places so as to meet exactly in the middle ?

Ques, 88.- Proposed by Mr. Dyer, Wanstead, A cylindrical glass, six inches long, is forced into a tub of water until the top of the glass is on a level with the water in the tub. How high will the water ascend in the glass ?

Ques. 89.- Proposed by Nemo. How many cubic feet of water must descend a river every minute to drive a water-wheel of P effective horse-powers, by means of a fall of h feet, the wheel yielding the nth part of the work of the fall ?

29

#ntelligence.

GOVERNMENT EXAMINATION If they are not in force, by what auFOR CERTIFICATES

OF MERIT thority and when have they been AND FOR QUEEN'S SCHOLARSHIPS. altered? Christmas, 1850.-We have ascer. 3. Is there any provision in your tained that about four hundred and present statutes for their alteration fifty candidates for certificates, and or amendment ? or was there, in upwards of fifty competitors for your original statutes, any such proQueen's Scholarships, attended the vision ? recent examination before H. M. In- 4. Will you state in what respects, spectors at the various Training Col- if any, your statutes have ceased to leges. The last-named class of can- be observed-whether owing to lapse didates are pupil teachers who have of time or other causes? completed the term of their appren- 5. In what case is the non-resiticeship, and are thus eligible to com- dence of your head or your fellows pete for the advantages held out to permitted by the statutes ? and how them under the minutes of 1846, many of your fellows are non-resi. consisting of exhibitions to the a- dents? Would the University or the mount of 201. or 251. towards de college be benefited, in your opinion, fraying the cost of their residence in by the general enforcement of resiany training school, to be selected dence ? by themselves, under certain limi. 6. Is the marriage of the head of tations.

your college permitted by the staWe subjoin the names of the In- tutes ? If not, by what authority is stitutions at which an examination such permission granted ? Is that has been held at Christmas, with the permission applicable to any other name of the Inspector by whom it members of the foundation besides was conducted in each case :- - Bat- the head ? tersea, Rev. J. J. Blandford; St. 7. Does the college consist of seveMark's, Chelsea, Rev. W. H. Brook- ral foundations? If so, do the fellows field; Home and Colonial, Rev. M. on the several foundations enjoy the Mitchell; Whitelands, Rev. F. C. same rights and advantages ? Cook ; Cheltenham, Rev. H. W. 8. Are there in your college any Bellairs ; Chester, Rev. J.P. Norris; unincorporated or by-fellows? If York, Rev. F. Watkins ; Durham, so, by what statutes are they goRev. D. J. Stewart; Salisbury, Rev. verned? Do you consider such felE. D. Tinling; Carmarthen, Rev. H. lowships beneficial to the society? L. Jones; Warrington, Rev. W. J. Or do you think their present posiKennedy; Kneller Hall, the Rev. F. tion might be altered with advanTemple, Principal; Borough Road, J. Fletcher, Esq.

9. How

many

of

your fellowships, OXFORD UNIVERSITY COMMIS. studentships, scholarships, exhibi. SION. – An additional circular has tions, or the like, are at present open been issued by Mr. Stanley, the Se- to competition without restriction ; cretary to the Oxford University and how many confined to particular Commission, requesting information places or schools, or to persons of on the following heads of inquiry :- the kin or name of founders ?

1. Is your society governed by 10. Will you quote the clauses of statutes ?" If not, are there any or- your statutes on which any such reders or rules by which it is go- striction rests, together with any verned ?

special reasons which the founder or 2. If the society is governed by framer of your statutes may have had statutes, were those statutes given for this restriction ? by the founder? Are the original 11. Is the restriction absolute? statutes in force, wholly or in part ? If not, has the college availed itself

tage ?

S

of any facilities which the statutes 21. Is the head of your society allow for opening the foundation ? statutably required to enter into holy 12. If the statutes give a “prefer- orders ? How many of

your

fellows, ence” to certain candidates, how do students, or the like, are subject to you interpret such preference ? the same rule? If the statute be not

13. Do you consider the present observed, on what authority does the restrictions, if any, on the election of non-observance or dispensation rest? your fellows, students, scholars, de- Is the obligation to enter into holy mies, or the like, to be beneficial to orders expressly laid down by stathe promotion of education or learn- tute, or is it deduced from an injuncing in the University, in your own tion to study theology, from an insociety, and in the particular place, junction to discharge clerical duties school, or family, if there be any, now disused, or from any other like which is supposed to be favoured by provision. those restrictions ?

22. Are your fellowships confined 14. Are your fellowships, student- to persons of a certain University ships, scholarships, demyships, or standing? Is the admission of unthe like, disposed of strictly accord- dergraduates to fellowships, or the ing to merit? Is such merit tested restriction of fellowships to persons by examinations?

of particular degrees, productive of 15. What is the statutable con- inconvenience ? nection between your fellowships and 23. Are clergymen excluded from your scholarships, demyships, or the presenting themselves as candidates like? What is the practice in this for your fellowships? Are laymen? respect?

24. Are fellows or other members 16. If

your fellowships are limited on your foundation allowed by stato those who are or have been scho- tute or other authority to hold ecclelars, or the like, of your society, has siastical preferment? and, if so, to this system been found beneficial to what amount? the college?

25. What statutable restrictions 17. Are the fellows of your college limit the selection of

your

head? obliged to proceed to the higher de- 26. How many benefices in the grees? If so, in what faculties ? gift of your society have been added

18. Do your statutes enjoin that since the original foundation ? Will your fellowships, studentships, scho- you state at what time and in what ·larships, or the like, be increased or manner each was acquired? Have diminished in number as the revenues you at present a fund for the purof your college vary? Has such pro- chase of advowsons ? vision of the statutes been acted 27. Are there any prælectorships upon? Do you conceive that the en. founded in your college for the beforcement of such provision at the nefit of the whole University ? Are present time would be beneficial to fellowships connected with such præthe society?

lectorships? If so, do the statutes 19. Do your statutes contemplate allow any special liberty of choice in the residence in college of any under- the fellowships so connected ? graduates not on the foundation ? 28. Has the college the nomination Do they forbid it? Will you quote to the masterships of any schools ? the clauses in your statutes

What control does the college exerwhich such permission or prohibi- cise over such schools ? tion rests ?

29. Will you quote the clauses of 20. What amount of property va- your statutes which set forth the cates a fellowship, scholarship, or powers and duties of the visitor of the like, according to your statutes ? your college? Has the visitor eyer Is the rule enforced equally with re- interposed his authority to relieve gard to real and personal property? the college from the observance of Do you conceive that the enforce- any of the statutes, or to make new ment of such a rule at the present statutes or ordinances ? time would be beneficial to the so. 30. Are gentlemen commoners in ciety?

your society called upon to pass the

on

same examination at entrance as society, are engaged as private tuother persons? Do they follow the tors ? same course of studies, and are they 39. Can you state how many undersubjected to the same discipline as graduate members of your society other persons in statu pupillari ? To are now reading with private tutors ? what charges are they liable beyond 40. What attendance at chapel is those borne by other independent required by your statutes ? What members ?

attendance is actually enforced? And 31. Can you state how many mem- by what means ? Is attendance at bers of your society receive assistance chapel ever enforced as a punishfrom exhibitions or the like, not in ment? the gift or under the administration 41. What is the nature and exof your society? What are the tent of religious instruction given in sources and what is the amount of your society, distinguishing lectures the assistance so received ?

and sermons delivered in chapel, and 32. How many persons are sup

instruction given in other ways ? ported, wholly or in part, in your 42. What is the average amount society as batellers, servitors, Bible- of the “battels” of each independent clerks, or the like? What are their member of your society? What was duties, and what are their sti ends or the

ighest and what was the lowest other emoluments or immunities? amount in the year 1849 ? How are they chosen? Are they 43. Will you have the goodness to marked by any particular dress? supply the commissioners with a Was the number ever greater ? If weekly battel bill of the average so, can you state why it has been re- amount, and with a quarterly battel duced ? What do you consider to be bill for each of the four quarters of the advantage or disadvantage of 1849, also of the average amount ? such a body of scholars ?

44. What is the lowest yearly sum 33. How many tutors are there in for which you have known an underyour society? How many lecturers, graduate to live in your society? catechists, or other instructors, who What is the lowest amount which are not tutors? Does the head of you have known an undergraduate your society take any direct part in to expend, from his matriculation to the instruction ?

his graduation ? 34. Are there any tutors in your 45. Do you conceive that the colsociety who are not, or have not been, lege expenses could be materially on the foundation ? Do they all re- diminished ? If so, will you state in side within the walls ?

what respects ? 35. Is each tutor expected to lec. 46. Is the college library open to ture on all subjects; or is there a all members of the college, and what division of subjects ?

fees are paid to the library by each 36. During how many weeks in member the year are lectures given in your 47. What number of under-grasociety? Will you state the average duates is your college capable of acnumber of lectures given weekly, commodating? and the subjects? How many undergraduates attend mathematical lec

The two following circulars have tures beyond arithmetic and the ele

been issued by the Commission to the ments of Euclid and algebra ?

Master and Fellows of each College, 37. Are any members of the college

and to the authorities of the Vicerequired to attend any professor's

Chancellor's Court:lectures ? And are any means adopted by the college to secure profitable

Oxford University Commission, attendance by examination or other- Downing-street, Nov. 1850. wise ?

“Gentlemen-Her Majesty's Com38. Can you state how many mem- missioners for the University of bers of your foundation, and how Oxford will feel much obliged for wany ndependent members of the any information which you may be

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