251. The curves of greatest inclination to the plane of (x, y) are determined by the given equation combined with x-y constant. 252. The curves are determined by the given equation combined with x2- y = constant.

The specified evolute is determined by the second equation combined with 27 a (y + z + 4a)2 = 2 (x − a)3.

1

238. x=0, y=-√(aß-ß3), z = {(a – ß); supposing a> ß.

299. The condition is abc + 2a'b'c' — aa2 – bb'2 - cc'2 = 0.

сс

302. Aa + BB + Cy=0, is the equation to the plane which is parallel to the face of which the area is D, and which passes through the opposite vertex.

305. The equation represents a cone which touches the planes represented by u ̧=О, u ̧=0, u ̧1⁄2=0.

306. The equation represents a cone containing the lines of intersection of the planes u1 = 0, u ̧=0, u ̧=0.

T. A. G.

6

313. An ellipsoid, a point, or an impossible locus according as a is >= or < <0.

314. An hyperboloid of one sheet, a cone, or an hyperboloid of two sheets according as a is >= or <

<0.

An elliptic cylinder, a straight line, or an impossible locus according as a is >= or< 0.

316. An hyperbolic cylinder, two planes, or an hyperbolic cylinder according as a is > = or < 0.

317. Two parallel planes, one plane, or an impossible locus according as a is > = or < 0.

318. An elliptic paraboloid.

boloid.

319. An hyperbolic para

320. A parabolic cylinder. See for the last eight

questions The Mathematician, Vol. I. p. 195.

In examples 321-329 the symbols r, 0, & are the usual polar co-ordinates. 321. A right circular cone having its vertex at the origin and its axis coincident with the axis of z. 322. A plane containing the axis of z. 323. A sphere having its centre at the origin. 324. A series of right circular cones having their vertices at the origin and their axes coincident with the axis of z. 325. A series of planes containing the axis of %. 326. A series of spheres having the origin for centre. surface of revolution round the axis of z. that any section made by a plane which

is a circle with the origin for centre.

327. A

328. A surface such contains the axis of z 329. A conical surface

generated by straight lines which all pass through the origin.

331. A sphere having the origin on its surface.

340. Let c be a side of the hexagon, a an edge of the cube;

c√3 then a = √2

342. Let a be the edge of the cube; the height of the luminous point above the given plane is a (2 + √2).

347. The figure formed by the revolution of an hyperbola round its conjugate axis.

352. (1 − m2) (≈2 — c3) = y3 — m2x2; the axes being as in Example 74.

354. The equation to the surface is

2cx2 +≈ {(x + y)3 − 2c3} - 4cxy = 0; the volume is

83πε 240/2

х

29

363. xa2 = ky12, and x2 + y2+z2 = log (k'y263).

2

380. (x2 + y2+)3 = 6c3xyz where c3 is the constant volume.

straight line, and +

+x2 + y2 = l2, where 27 is the length of the

=

a2 b2

405. The longest axis of the ellipsoid must be inclined to the

plane at an angle whose cosine is

2

406._y3 {p (≈)}2 = {x − 4 (≈)}3 [c3 — {6 (≈)}3].

408. æ {a (a + b + c − a (ca +BB + y)} + ... = 0.

409. by-cẞ Example 45.

= &c.; co-ordinates those given in the result of

410. Let a be the edge, p the distance of the

section from one corner. Then from p = 0 to p=

α

√3

the section is

p23/3

an equilateral triangle; the perimeter is 3p/6 and the area 2

sides equal and also the other three sides equal; the perimeter is 3p √6-3√2 (p√3 − a), that is 3a/2, and the area is · √3, p$3/3 3 {√2 (p √3−a)}2 2

From p

2a

9 4

that is 9ap-3p2 √3

3a2 √3 2

to pa√3 the results may be obtained from

those in the first case by putting a 3-p instead of p. The area

a√3

is greatest when p= 2

formed by revolving a circle about a straight line in its plane.

CAMBRIDGE: PRINTED AT THE UNIVERSITY PRESS.

AN ELEMENTARY LATIN GRAMMAR.
BY H. J. ROBY, M.A.

Under Master of Dulwich College Upper School, late Fellow and Classical
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THE Author's experience in practical teaching has induced him to treat Latin Grammar in a more precise and intelligible way than has been usual in school books. The facts have been derived from the best authorities, especially Madvig's Grammar and other works. The works also of Lachmann, Ritschl, Key, and others have been consulted on special points. The accidence and prosody have been simplified and restricted to what is really required by boys. In the Syntax an analysis of sentences has been given, and the uses of the different cases, tenses and moods briefly but carefully described. Particular attention has been paid to a classification of the uses of the subjunctive mood, to the prepositions, the oratio obliqua, and such sentences as are introduced by the English 'that.' Appendices treat of the Latin forms of Greek nouns, abbreviations, dates, money, &c. The Grammar is written in English.

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MYTHOLOGY FOR VERSIFICATION.

A Brief Sketch of the Fables of the Ancients, prepared to be rendered into Latin Verse for Schools.

BY THE REV. F. HODGSON, B.D.

Late Provost of Eton College.

New Edition, revised by F. C. HODGSON, M.A. Fellow of King's College, Cambridge. 18mo. 38.

The Author here offers to those who are engaged in Classical Education a further help to the composition of Latin Verse, combined with a brief introduction to an essential part of the study of the Classics. The Author has made it as easy as he could so that a boy may get rapidly through these preparatory exercises and thus having mastered the first difficulties, he may advance with better hopes of improvement to subjects of higher character and verses of more difficult composition.