Mathematical Questions and Solutions

LIST OF CONTRIBUTORS.

ALDIS, J. S., M.A.; H.M. Inspector of Schools. | EVANS, Professor, M.A.; Lockport, New York.

ALLEN, Rev. A.J.C., M.A.; St. Peter's Coll., Camb.
ALLMAN, Professor GEO. J., LL.D.; Galway.
ANDERSON, ALEX., B.A.; Queen's Coll., Galway.
ANTHONY, EDWYN, M.A.; The Elms, Hereford.
ARMENANTE, Professor; Pesaro.
BALL, ROBT. STAWELL, LL.D., F.R.S.; Professor
of Astronomy in the University of Dublin.
BASU, SATISH CHANDRA; Presid. Coll.,Calcutta.
BATTAGLINI, GIUSEPPE Professore di Mate-
matiche nell' Università di Roma.
BAYLISS, GEORGE, M.A.; Kenilworth.
BELTRAMI, Professor; University of Pisa.
BERG, F. J. VAN DEN; Professor of Mathematics
in the Polytechnic School, Delft.
BESANT, W. H., D.Sc., F.R.S.; Cambridge.
BHUT, ÁTH BIGAH, M.A.; Delhi.
BICKERDIKE, C.; Allerton Bywater.

BIDDLE, D.; Gough H., Kingston-on-Thames.
BIRCH, Rev. J. G., M.A.; London.
BLACKWOOD, ELIZABETH; Boulogne.
BLYTHE, W. H., B.A.; Egham.
BORCHARDT, Dr. C. W.; Victoria Strasse, Berlin.
BOSANQUET, R. H. M., M.A.; Fellow of St.
John's College, Oxford.

BOURNE, C. W., M.A.; Bedford County School.
BROOKS, Professor E.; Millersville, Pennsylvania.
BROWN, A. CRUM, D.Sc.; Edinburgh.
BROWN, Prof.COLIN; Andersonian Univ.Glasgow.
BUCHHEIM, A.,M.A., Ph.D.; Schol. New Coll.,Oxf.
BUCK, EDWARD, M.A.; Univ. Coll., Bristol.
BURNSIDE, W. S., M.A.; Professor of Mathe-
matics in the University of Dublin.
CAPEL, H. N., LL.B.; Bedford Square, London.
CARMODY, W. P., B.A.; Clonmel Gram. School.
CARR, G. S., B.A.; Endsleigh Gardens, London.
CASEY, JOHN, LL.D., F.R.S.; Prof. of Higher
Mathematics in the Catholic Univ. of Ireland.
CAVALLIN, Prof., M.A.; University of Upsala.
CAVE, A. W., B.A.; Magdalen College, Oxford.
CAYLEY, A., F.R.S.; Sadlerian Professor of Ma-
thematics in the University of Cambridge,
Member of the Institute of France, &c.
CHAKRAVARTI, BYOMAKESA, M.A.; Professor
of Mathematics, Calcutta.

CHASE, PLINY EARLE, LL.D.; Professor of Phi-
losophy in Haverford College.
CLARKE, Colonel A. R., C.B., F.R.S.; Hastings.
COATES, W. M., B.A.: Trinity College, Dublin.
COCHEZ, Professor; Paris.

COCKLE, Sir JAMES, M.A., F.R.S.; Bayswater.
COHEN, ARTHUR, M.A., Q.C., M.P.; Holland Pk.
COLSON, C. G., M.A.; University of St. Andrew's.
CONSTABLE, S.; Swinford Rectory, Mayo.
COTTERILL, J. H., M.A.; Royal School of Naval
Architecture, South Kensington.
CREMONA, LUIGI; Direttore della Scuola degli
Ingegneri, S. Pietro in Vincoli, Rome.
CROFTON, M. W., B.A., F.R.S.; Prof. of Math.
and Mech. in the R. M. Acad., Woolwich.
CULVERWELL, E.P.,B.A.; Sch. of Trin. Coll..Dubl.
CURTIS, ARTHUR HILL, LL.D., D.Sc.; Dublin.
DARBOUX, Professor; Paris.
DAVIS, J. G., M.A.; Abingdon
DAVIS, R. F., B.A.; Wandsworth Common.
DAWSON, H. G., B.A.; Baymount, Dublin.
DAY, Rev. H. G.,M.A.; Richmond Terr., Brighton.
DEY, Prof. NARENDRA LAL, M.A.; Calcutta.
DICK, G. R., M.A.; Fellow of Caius Coll., Camb.
DOBSON, T., B.A.; Hexham Grammar School.
DROZ, Prof. ARNOLD, M.A.; Porrentruy, Berne.
DUPAIN, J.C.; Professeur au Lycée d'Angoulême.
EASTERBY, W., B.A.; Grammar School, St. Asaph.
EASTWOOD, G., M.A.; Saxonville, Massachusetts.
EASTON, BELLE; Lockport, New York.
EDWARD, J., M.A.; Head Mast., Aberdeen Coll.
EDWARDES, DAVID; Erith Villas, Erith, Kent.
ELLIOTT, E. B., M.A.; Fell. Queen's Coll., Oxon.
ELLIS, ALEXANDER J., F.R.S.; Kensington.
EMTAGE, W. T. A.; Pembroke Coll., Oxford.
ESCOTT, ALBERT, M.A.; Head Master of the
Royal Hospital School, Greenwich.
ESSENNELL, EMMA; Coventry.

FICKLIN, JOSEPH; Prof. in Univ. of Missouri.
FINCH, T. H., B.A.; Trinity College, Dublin.
FORTEY, H., M.A.; Bellary, Madras Presidency.
FOSTER, F. W., B.A.; Chelsea.
FOSTER, Prof.G.CAREY, F.R.S.; Univ. Coll., Lond.
FRANKLIN, CHRISTINE LADD, M.A.; Prof. of Nat.
Sci. and Math., Union Springs, New York.
FUORTES, E.; University of Naples.
GALBRAITH,Rev.J.,M.A., Fell.Trin.Coll., Dublin.
GALE, KATE K.; Worcester Park, Surrey.
GALLATLY, W., B.A.; Earl's Court, London.
GALLIERS, Rev. I., M.A.; Kirkstead Rectory,
Norwich.

GALTON, FRANCIS, M.A., F.R.G.S.; London.
GENESE, Prof., M.A.; Univ. Coll., Aberystwith.
GERRANS, H. T., B.A.; Stud. of Ch. Ch., Oxford.
GLAISHER, J. W. L., M.A., F.R.S.; Fellow of
Trinity College, Cambridge.

GOLDENBERG, Professor, M.A.; Moscow.
GRAHAM, R. A., M.A.; Trinity College, Dublin.
GREENFIELD, Rev. W. J., M.A.; Dulwich College.
GREENSTREET, W. J., B.A.; Framlingham.
GREENWOOD, JAMES M.; Kirksville, Missouri.
GRIFFITH, W.; Superintendent of Public Schools,
New London, Ohio, United States.
GRIFFITHS, G. J., M.A.; Fell. Ch. Coll., Camh.
GRIFFITHS, J., M.A.; Fellow of Jesus Coll., Oxon.
GROVE, W. B., B.A.; Perry Bar, Birmingham.
HADAMARD, Professor, M.A.; Paris.
HAIGH, E., B.A., B.Sc.; King's Sch., Warwick.
HALL, Professor ASAPH, M.A.; Washington.
HAMMOND, J., M.A.; Buckhurst Hill, Essex.
HARKEMA, C.; University of St. Petersburg.
HARLEY, Harold, B.A.; King's Coll., Cambridge.
HARLEY, Rev. R., F.R.S.; Huddersfield College.
HARRIS, H. W., B.A.; Trinity College, Dublin.
HARRIS, J. R., M.A.; Clare College, Cambridge.
HART, Dr. DAVID S.; Stonington, Connecticut.
HART, H.; R.M. Academy, Woolwich.
HAUGHTON, Rev. Dr., F.R.S.; Trin. Coll., Dubl.
HENDRICKS, J. E., M.A.; Des Moines, Iowa.
HEPPEL, G., M.A.; The Grove, Hammersmith.
HERBERT, A.,M.A.; King Alfred's Sch., Wantage.
HERMAN, R. A., M.A.; Trin. Coll., Cambridge.
HERMITE, CH.; Membre de l'Institut, Paris.
HILL, Rev. E., M.A.; St. John's College, Camb.
HINTON, C. H., M.A.; Cheltenham College.
HIRST, Dr. T. A., F.R.S.; Director of Studies in
the Royal Naval College, Greenwich.
HOLT, J. R., M.A.; Trinity College, Dublin.
HOPKINS, Rev. G. H., M.A.; Stratton, Cornwall.
HOPKINSON, J., D.Sc., B.A.; Kensington.
HUDSON, C. T., LL.D.; Manilla Hall, Clifton.
HUDSON, W.H.H.,M.A.;Prof.in King's Coll., Lond.
INGLEBY, C. M., M.A., LL.D.; London.
JENKINS, MORGAN, M.A.; London.
JOHNSON, W. E., B.A.; King's Coll., Cambridge.
JOHNSON, Prof., M.A.; Annapolis, Maryland.
JOHNSTON, SWIFT; Trin. Coll., Dublin.
JONES, H. S., M.A.; Llanelly.
JONES, L. W., B.A.; Merton College, Oxford.
KEALY, J. A., M.A.; Wilmington, Delaware.
KENNEDY, D., M.A.; Catholic Univ., Dublin.
KIRKMAN, Rev. T. P., M.A., F.R.S.; Croft Rect.
KITCHIN, Rev. J. L., M.A.; Heavitree, Exeter.
KITTUDGE, LIZZIE A.; Boston, United States.
KNISELY, Rev. U.J.; Newcomerstown, Ohio.
KNOWLES, R., B.A., L.C.P.; Tottenham.
KOEHLER, J.; Rue St. Jacques, Paris.
LACHLAN, R., B.A.; Lewisham.
LARMOR, J., M.A.; Queen's College, Galway.
LAVERTY, W. H., M.A.; Public Examiner in the
University of Oxford.

LAWRENCE, E. J.; Ex-Fell. Trin. Coll., Camb.
LAX, W. G., B.A.; Trinity College, Cambridge.
LEIDHOLD, R., M.A.; Finsbury Park.
LEUDESDORF, C., M.A.; Fel. PembrokeColl.,Oxon.
LEVETT, R., M.A.; King Edw. Sch., Birmingham.
LOWRY, W. H., M.A.; Blackrock, Dublin.
MACDONALD, W. J., M.A.; Edinburgh.
MACFARLANE, A., D.Sc., F.R.S.E.; Ontario.
MACKENZIE, J. L., B.A.; Gymnasium, Aberdeen.

EVERETT, J.D., D.C.L.; Professor of Natural Phi- MACMAHON, Capt. P. A.; R. M. Academy.

losophy in Queen's College, Belfast.

MACMURCHY, A., B.A.; Univ. Coll., Toronto.

MCALISTER, DONALD, M.A., D.Sc.; Cambridge. | SALMON, Rev. G., D.D., F.R.S.; Regius Professor
MCCAY, W. S., M.A.; Fellow and Tutor of

Trinity College, Dublin.
MCCLELLAND, W. J., B.A.; Prin. of Santry School.
MCCOLL, H., B.A.; 73, Rue Sibliquin, Boulogne.
MCDOWELL, J., M.A.; Pembroke Coll., Camb.
MCINTOSH, ALEX., B.A.; Bedford Row, London.
MCLEOD, J., M.A.; R.M. Academy, Woolwich.
MCVICKER, C. E., B.A.; Trinity Coll., Dublin.
MALET, Prof., M.A.; Queen's Coll., Cork.
MANNHEIM, M.; Prof. à l'Ecole Polytech., Paris.
MARKS, SARAH; Cambridge Street, Hyde Park.
MARTIN, ARTEMAS, M.A., Ph.D.; Editor & Prin-
ter of Math. Visitor & Math. Mag., Erie, Pa.
MARTIN, Rev. H., D.D., M.A.; Edinburgh.
MATHEWS, G. B., B.A.; Univ. Coll., N. Wales.
MATZ, Prof., M.A.; King's Mountain, Carolina.
MEE, W. M., B.A.; Belturbet.

MERRIFIELD, J., LL.D., F.R.A.S.; Plymouth.
MERRIMAN, MANSFIELD, M.A.; Yale College.
MEYER, MARY S.; Girton College, Cambridge.
C., B.A., (EDITOR);
MILLER,
The Paragon, Richmond-on-Thames.
MINCHIN, G.M.,M.A.; Prof. in Cooper's Hill Coll.
MITCHESON, T.,B.A.,L.C.P.; City of London Sch.
MONCK, HENRY STANLEY, M.A.; Prof. of Moral
Philosophy in the University of Dublin.
MONCOURT, Professor; Paris.
MOON, ROBERT, M.A.; Ex-Fell. Qu. Coll., Camb.
MOORE, H. K., B.A.; Trin. Coll., Dublin.
MOREL, Professor; Paris.

MORGAN, C., B.A.; Salisbury School.
MORLEY, FRANK, B.A.; Bath Coll., Bath.
MORLEY, T., L.C.P.; Bromley, Kent.
MORRICE, G. G., B.A.; The Hall, Salisbury.
MOULTON, J. F., M.A.; Fell. of Ch. Coll., Camb.
MUIR, THOMAS, M.A., F.R.S.E.; Bishopton.
MUKHOPADHYAY,ASUTOSH, M.A.; Bhowanipore.
NASH, A. M., M.A.; Prof. in Pres. Coll., Calcutta.
NELSON, R. J., M.Á.; Naval School, London.
NEWCOMB, Prof. SIMON, M.A.; Washington.
NICOLLS, W., B.A.; St. Peter's Coll., Camb.
OPENSHAW, Rev. T. W., M.A.; Clifton.
O'REGAN, JOHN; New Street, Limerick.
ORCHARD, H. L., M.A., L.C.P.; Burnham.
OWEN, J. A., B.Sc.; Tennyson St., Liverpool.
PANTON, A. W., M.A.; Fell. of Trin. Coll., Dublin.
PENDLEBURY, Rev. C., M.A.; London.
PERRYMAN, W.; Carbrook, Sheffield.
PHILLIPS, F. B. W.; Balliol College, Oxford.
PILLAI, C. K., M.A.; Trichy, Madras.
PIRIE, A., M.A.; University of St. Andrew's.
POLIGNAC, Prince CAMILLE DE; Paris.
POLLEXFEN, H., B.A.; Windermere College.
POTTER, J., B.A.; Richmond-on-Thames.
PRUDDEN, FRANCES E.; Lockport, New York.
PURSER, Prof. F., M.A.; Queen's College, Belfast.
PUTNAM, K. S., M.A.; Rome, New York.
RAU, B. HANUMANTA, B.A.; Head Master of
the Normal School, Madras.
RAWSON, ROBERT; Havant, Hants.
RAYMOND, E. LANCELOT, B.A., Surbiton.
RAY, Prof. SARADARANJAN, M.A.; Dacca Coll.,
Bengal.

READ, H. T., M.A.; Brasenose Coll., Oxford.
REEVES, G. M., M.A.; Lee, Kent.
RENSHAW, S. A.; Nottingham.

REYNOLDS, B., M.A.; Notting Hill, London.
RICHARDSON, Rev. G., M.A.; Winchester.
RIPPIN, CHARLES R., M.A.; Woolwich Common.
ROBERTS, R. A., M.A.; Schol.of Trin.Coll.,Dublin.
ROBERTS, S.,M.A., F.R.S.; Tufnell Park, London.
ROBERTS, Rev. W., M.A.; Senior Fellow of
Trinity College, Dublin.
ROBERTS, W. R.,M.A.; Ex-Sch. of Trin. Coll., Dub.
ROBSON, H. C., B.A.; Sidney Sussex Coll., Cam.
ROSENTHAL, L. H.; Scholar of Trin. Coll., Dublin.
ROY, Prof. KALIPRASANNA, M.A.; St. John's
Coll., Agra.

RÜCKER, A. W., B.A.; Professor of Mathematics

in the Yorkshire College of Science, Leeds.
RUGGERO, SIMONELLI; Università di Roma.
RUSSELL, J. W., M.A.; Merton Coll., Oxford.
RUSSELL, R., B.A.; Trinity College, Dublin.
RUTTER, EDWARD; Sunderland.

of Divinity in the University of Dublin.
SAMPSON, C. H., M.A.; Balliol College, Oxford.
SANDERS, J. B.; Bloomington, Indiana.
SANDERSON, Rev. T. J., M.A.; Royston, Cambs.
SARKAR, NILKANTHA, M.A.; Calcutta.
SAVAGE, THOMAS, M.A.; Fell. Pemb. Coll., Camb.
SCHEFFER, Professor; Mercersbury Coll., Pa.
SCOTT, A. W., M.A.; St. David's Coll., Lampeter.
SCOTT, CHARLOTTE A., B.Sc.; Manchester.
SCOTT, R. F., M.A.; Fell. St. John's Coll., Camb.
SERRET, Professor; Paris.
SHARP, W. J. C., M.A.; Hill Street, London.
SHARPE, J. W., M.A.; The Charterhouse.
SHARPE, Rev. H. T., M.A.; Cherry Marham.
SHEPHERD, Rev. A. J. P., B.A.; Fellow of
Queen's College, Oxford.

SIMMONS, Rev. T.C., M.A.; Christ's Coll., Brecon.
SIVERLY, WALTER; Oil City, Pennsylvania.
SMITH, C., M.A.; Sidney Sussex Coll., Camb.
STABENOW, H., M.A.; New York.

STEGGALL, Pro". J. E. A., M.A.; Clifton.
STEIN, A.; Venice.

STEPHEN, ST. JOHN, B.A.; Caius Coll., Cambridge.
STEWART, H., M.A.; Framlingham, Suffolk.
STORR, G. G., B.A.; Blackburn.
SWIFT, C. A., B.A.; Grammar Sch., Weybridge.
SYLVESTER, J. J., D.C.L., F.R.S.; Professor of

Mathematics in the University of Oxford,
Member of the Institute of France, &c.
SYMONS, E. W., M.A.; Fell. St. John's Coll., Oxon.
TAIT, P. G., M.A.; Professor of Natural Philo-
'sophy in the University of Edinburgh.
TANNER, Prof. H.W.L.,M.A.; S. Wales Univ. Coll.
TARLETON, F. A., M.A.; Fell. Trin. Coll., Dub.
TAYLOR, Rev. C., D.D.; Master of St. John's
College, Cambridge.

TAYLOR, H. M., M.A.; Fell. Trin. Coll., Camb.
TAYLOR, W. W., M.A.; Ripon Grammar School.
TEBAY, SEPTIMUS, B.A.; Farnworth, Bolton.
TERRY, Rev. T. R., M.A., Fell. Magd. Coll., Oxon.
THOMAS, Rev.D., M.A.; Garsington Rect., Oxford.
THOMSON, Rev.F.D.,M.A.; Ex-Fellowof St.John's

Coll., Camb.; Brinkley Rectory, Newmarket.
TIRELLI, Dr. FRANCESCO; Univ. di Roma.
TODHUNTER, ISAAC, F.R.S.; Cambridge.
TORELLI, GABRIEL; University of Naples.
TORRY, Rev. A. F., M.A.; St. John's Coll., Camb.
TRAILL, ANTHONY, M.A., M.D.; Fellow and
Tutor of Trinity College, Dublin.

TROWBRIDGE, DAVID; Waterburgh, New York.
TUCKER, R., M.A.; Mathematical Master in Uni-
versity College School, London.
TURRELL, I. H.; Cumminsville, Ohio.
TURRIFF, GEORGE, M.A.; Aberdeen.
VINCENZO, JACOBINI; Università di Roma.
VOSE, G. B.; Professor of Mechanics and Civil
Engineering, Washington, United States.
WALENN, W. H.; Mem. Phys. Society, London.
WALKER, J. J., M.A., F.R.S.; Hampstead.
WALMSLEY, J., B.A.; Eccles, Manchester.
WARBURTON-WHITE, R., B.A., Salisbury.
WARREN, R., M.A.; Trinity College, Dublin.
WATHERSTON, Rev. A. L., M.A.; Bowdon.
WATSON, Rev. H. W.; Ex-Fell. Trin. Coll., Camb.
WERTSCH, Fr.; Weimar.

WHITE, J. R., B.A.; Worcester Coll., Oxford.
WHITE, Rev. J., M.A.; Cowley College, Oxford.
WHITESIDE, G., M.A.; Eccleston, Lancashire.
WHITWORTH, Rev. W. A., M.A.; Fellow of St.

John's Coll., Camb.; Hammersmith.
WICKERSHAM, D.; Clinton Co., Ohio.
WILKINS, W.; Scholar of Trin. Coll., Dublin.
WILLIAMSON, B., M.A.; Fellow and Tutor of
Trinity College, Dublin.

WILSON, J. M., M.A.; Head-master, Clifton Coll.
WILSON, Rev. J., M.A.; Rect. Bannockburn Acad.
WILSON, Rev. J. R., M.A.; Royston, Cambs.
WOODCOCK, T., B.A.; Wellington, Salop.
WOLSTENHOLME, Rev. J., M.A., Sc.D.; Professor
of Mathematics in Cooper's Hill College.
WOOLHOUSE, W. S. B., F.R.A.S., &c.; London
WRIGHT, Dr. S. H., M.A.; Penn Yan, New York.
WRIGHT, W. E., B.A.; Herne Hill.

YOUNG, JOHN, B.A.; Academy, Londonderry.

CONTENTS.

Mathematical Papers, &c.

Note on the Symmedian-Point Axis of a System of Triangles.

(R. Tucker, M.A.)..

Note on Biot's Formula. (Âsûtosh Mukhopadhyây.)

Note on the Solutions of Question 5672.....

A New Method of deriving Legendre's Formula

(Professor Cavallin, M.A.)

1192. (The Editor.) — In order to ascertain the heights of two

balloons (Q, M), their angles of elevation as set forth hereunder are ob-

served, at the same instant, from three stations (A, B, C) on the hori-

zontal plane, whose distances apart are AB = 553, BC

yards, (Q, A) denoting the elevation of Q at A, &c. :-

791, CA

399

It is also observed that only one of the balloons (Q) is vertically over

the triangle ABC. Show that the heights of the balloons Q, M are

1874-8, 3339-4, and that their distance apart is 1560 4......

1208. (The Editor.) - Show that the values of x, y, z, from the

equations

x2 + 4xy+6y2 = 28, x2+4xz + 14z2 60, 3y2+2yz + 722 = 40...(1, 2, 3),

are given by

x2 =

(±√5-1) (±5√√/2-6) (±√/10-2),

y2 = (±√√/5+1) (±5.√√/2−6) (±√√/10+2),

22(±√5+1) (±5√√/2+6) (±√/10-2).

..........

122

1966. (The late Samuel Bills.)-Find values of x, y that will make

S=(p2+q2)+64 p2q2 ( p2 — q2)2 a perfect square.....

3826. (J. B. Sanders.)-The heights of the ridge and eaves of a

house are 40 feet and 32 feet respectively, and the roof is inclined at 30°

to the horizon. Find where a sphere rolling down the roof from the ridge

will strike the ground, and also the time of descent from the eaves......116

4038. (Rev. T. P. Kirkman, M.A., F.R.S.)-Prove that (1) a tri-
angle can be partitioned into 13 triangles in 457 ways, two ways being
reckoned the same if one is in any position the reflected image of the
other, the size of the partitions being of no consequence; and find (2) in
how many ways an equilateral triangle can be partitioned into 13
triangles of equal area..

108

4165. (J. Conwill.)-A fish is floating in a cubical glass tank filled

with water, with its head in one corner, and its tail towards the one

diagonally opposite; describe the appearance which will be presented to

an eye looking towards the corner in the direction of the length of the

fish, and in the same horizontal plane with it.

117

4390. (The Editor.) - Two gamesters, A and B, play together,

A having the power to fix the stakes. Whenever A loses a game, he

increases the last stake by a shilling for the next game, and diminishes it

by a shilling after every gain. When they leave off playing, A has

gained £13; and, had each won the same number of games, A would

still, by following the above principle in regulating his stakes, have

gained 10s. If the first stake be 30s., show that A won 15 and lost 5

games.....

4737. (Professor Artemas Martin, M.A., Ph.D.) — Three equal
circles, each 4 inches in diameter, are drawn at random on a circular
slate whose diameter is 12 inches; find the probability that each circle
intersects the other two......

......

118

5200. (S. Tebay, B.A.)--A small marble is thrown at random on

a square table having an elevated rim. If it be struck at random

in any direction, determine the probability that it impinges (1) on two

opposite sides; (2) on two adjacent sides, and one opposite; (3) on three

consecutive sides; (4) on the four sides in succession.

5218. (The Editor.)-A circular target revolves uniformly around

a vertical axis, lying in its plane and passing through its centre; and a

shot is fired at the target in (1) a given or (2) a random direction: find,

in the first case, the chance that the shot will hit the target, and show

therefrom that, in the second case, the chance is 2/π.

109

5481. (Professor Burnside, M.A.)-Trace the relation between the
characteristics of a curve of the mth degree having the maximum number
of double points, and the curve enveloped by the line

...

(a0, a1, a2, am) (0, 1)m

...

am are linear functions of the coordinates, and a

where do, a1, a2,
variable parameter.
5501. (Professor Ball, LL.D., F.R.S.)-If in an equation x be changed
into +x-1, show that any semi-invariant of the transformed will be a
covariant in k of the original equation.........
67

5635. (Elizabeth Blackwood.)-Two excursion trains, each m yards
in length, may start with equal probability from their respective stations
at any time between 2 o'clock and 10 minutes past 2, in directions
at right angles to each other, each at a uniform rate v; find the chances
of a collision, each being n yards distant from the point at which their
lines cross, and both being ignorant of the risk they are running.. 36
5636. (C. Leudesdorf, M.A.)-A polished uniform straight metal rod
is held in a horizontal position with one end fixed at a point A, and is

......

then allowed to swing under the action of gravity till it reaches a vertical

position, when the end A is loosed, and the rod allowed to fall; find the

locus traced out by the image of the fixed point A, as seen from any point

by reflection at the rod during the motion of the latter.

5672. (Col. Clarke, C.B., F.R.S.) P and Q are two points in a
finite line AB. The parts PA, QB are rotated in opposite directions
round P and Q respectively, until A and B meet in a point R. Supposing
P and Q evenly distributed, determine the law of density of the point R.

6118. (Professor Sylvester, F.R.S.)-A plane or solid reticulation,

rigid but without weight, is formed by the intersections of equi-distant

lines or planes. One of these intersections is fixed, and at a certain

number of others of them, which are given, forces may be applied.

It is obvious that there are an infinite number of sets of parallel forces

each containing an exact number of pound weights, which, acting at the

given points of application, will balance about the fixed point.

It is required to prove that out of these a limited number may be

selected such that by their due repetition and superposition any other

balancing set whatever may be formed. In other words, i balancing sets of

parallel forces P, Q... W (i being some finite number) may be found such

that any other balancing set will be made up of m1 of the first set or