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wherefore this last right line, as being the common axis of a series of such right cylinders, may be called the Central Axis of the System. 6. The equation,
SQc = c = any scalar constant, represents a plane; and all such planes are parallel to the Central Plane, of which the equation is,
SQc=o. 7. Prove that the central axis intersects the central plane perpendicularly, in the central point of the system.
8. When the n given vectors A1B1, .. AnBn are parallel, and are therefore proportional to n scalars, bi, .. bn, the scalar h and the vector Qg vanish; and the centre H is then determined by the equation,
b. HA1 +62. HA2+.. tbn. Han= 0, or by the expression,
b. 041+ + bn. O An OH
bit..tbno where 0 is again an arbitrary origin.
DIFFERENTIAL EQUATIONS, AND ENVELOPES IN SPACE. 1. If 0, 4, a be regarded as arbitrary functions of a variable a, the two equations,
(a). (x -0)2+(y-7)+(2-7)2 = a?,
(x - 0) 0'+(y-7)+(2-7)+'+a=0, represent, by elimination of a, the envelope of a sphere of which the centre out traverses an arbitrary curve in space, while its radius a varies, as has been shown by Mongea, who also writes for abridgment,
(c)... 1-0'2 - 4'2 – 7"? = h. Prove that when Monge's quantity h is real, his envelope (a) (6) is imaginary; and that conversely his radical h is imaginary, precisely when that envelope (a) (b) of a sphere, with one varying parameter, is real.
2. When h=0, the generally superficial envelope (a) (6) reduces itself to a curve, as its only real part, with the locus of the centre for an evolute.
3. When h2 <o, the two equations (a) (b), for any given and real value of a, represent a real circle, of which the real envelope of the sphere is the locus; and all such circles touch two generally distinct cusp-edges, or two branches of one cusp-edge, determined by the system of the three equations (a) (6) and
(d) ... (2-0) " +(y-44"+(2-7) 7" + h = 0, with the value (c) of ha.
a Liouville's Edition, pages 369, &c.; or Fourth Edition (1809), pages 322, &c. An exponent, missing in both Editions, has been restored in the equation (e).
4. The co-ordinates xyz of a point of such a cusp-edge (a) () (d) contain a common radical, by equating which to zero Monge obtains the equation of condition,
(e)... [a (0'0" + 4'4" + o'rt") – h2]2 +h2 [a? (0"2 +\"2+ 7"2) – h4] = 0, as a differential equation of the second order and second degree, which must be satisfied by the three functions 0, 4, 7 of a, in order that the two branches of the cusp-edge may coincide, or be confounded into one curve, which has then no longer a cuspidal character, but is still a singular line on the surface, which is as before the welope of the sphere. Without actually calculating the radical last mentioned, show by geometrical considerations that Monge's equation (e) must admit of being otherwise obtained, as the result of the elimination of xyz between the three equations (a) (b) (d), and the following, ) ... (1-0) (UT" - 7'4")+(-) 1'0" - O'r")
+(3-6"-49") = 0, which does not seem to have been noticed by him.
5. By treating xyz as functions of a, the four equations (a) (6) (d) (5) conduct to the three following,
() ... (1-0)*' +(y-7) y' + (-7)' = 0,
(0) ... (x – °) x" + (9-4) y'" +(2–7) z*" +3 (cʻx" +y'y" +zz") = 0; auxiliary relations being,
(j) a'd' + y't' + 'n' = 0,
y"f' +="ht' = 0; while (d) and (k) give also, (m)... (2-0) "+(y-7)+""+(z-7)+"- 3 ('9" + '"+") = 0, How does the new equation (f) assist in the deduction of (k) ?
6. Conversely, the four equations (a) (6) (d) (m) give the three equations (9) (1) (k); and therefore, if we do not suppose that xyz are the coordinates abc of a fixed point, they conduct by elimination of x'y'z' to the equation (f), and so to Monge's equation of condition (e); which thus expresses, generally, that four consecutive spheres of the series intersect in a common point xyz, namely in the point of the singular line (4).
7. Hence also, or from the four equations (a) (g) (h) (i), it may be inferred that any one sphere (a) of the series passes through four consecutive points of the same singular line; or in other words, that it is the osculating sphere to that singular line (4) at the point xyz.
8. The circle (a) (6) is therefore the osculating circle to the same line, at the same point; and accordingly the equations (j) (?) express, that the tangent to the curve (1) is perpendicular to the osculating plane to the singular line, as it ought to be, because that curve is by (7) the locus of the centres of the osculating spheres to that line (4), or the cusp-edge of its polar developable.
9. Conversely, the tangent to the singular line is perpendicular, by (;) and (k), to the osculating plane to the curve, in which plane by (f) the point xyz is situated; the line therefore crosses that plane at that point
perpendicularly: or, in other words, the singular line (4) of the envelope is an orthogonal trajectory to the osculating planes of the proposed curve (1), which curve is therefore seen again to be the cusp-edge of its polar developable.
10. If we change a in (6) to aa', to represent that the radius a, and the co-ordinates of the centre of the sphere, are conceived, for greater generality, to be all functions of an independent variable t, we are then to change the term +h2 in (d) to such an expression as the following,
(n)... +e=aa'+a’– 02 – 4'2 - a's; and then if we write, for abridgment,
( )... a=9'3+42 + 7"?, b = 0' 9" +4'4" +a'r", o=0" +42 +77", Monge's equation of condition (e) takes this more general form,
(p)... (ba? – eaa')2 = (aa: - a?a’2) (ca? – (*); namely that of a differential equation, of the second order and second degree, wherein four variables, , 4, , and a, are regarded as functions of a fifth.
11. The complete and general integral of this last equation (p) is expressed by the system of the four equations (a) (9) (h) (i), in which the three variables x, y, z are treated as arbitrary functions of the same independent variable t; because we can return from this last system to (6) (1)(), and thence to (d) and (k), with the new forms of (b) and (d); while (9) (j) (k) oonduct, as before, by elimination of the ratios of x'y'z, to (f); and thence, by elimination of xyz to the extended form (p) of Monge's equation (e).
12. In this integral system (a) (g) (h) (i), it is permitted to assume any variable, for example, "z, instead of t, as that with respect to which the differentiations are to be performed, and so to write
a = 1, 2"=0, 2"= 0; the complete and general integral of the original equation (e) depends therefore, in the last analysis, on only two arbitrary functions, such as those which determine the form and position of the line (4), considered now as an arbitrary curve in space, from which the co-ordinates 9,4, a of the curve (1) and the radius a can be deduced, by the formulæ for an osculating sphere (7).
13. A less general integral can however be found, which involves only one arbitrary function, corresponding to the case where the curve (1) is a plane one, and the spheres are all obliged to pass through a fixed point A in its plane: in which particular case, the singular line (4) reduces itself to the singular point A.
UNDERGRADUATE HONOR EXAMINATION PAPERS.
EXAMINATION FOR THE DEGREE OF BACHELOR OF
Moderatorships in Mathematics and Mathematical Physics.
1. If several curves of the nth degree pass through the same na points, prove that the tangents at these points form homographic pencils.
2. Find the condition that the line la + mß + nyro should touch a given conic.
3. If a chord of a conic pass through a fixed point, prove that the product of the tangents of the halves of the angles which its segments subtend at the focus is constant.
4. Prove that every cubic which passes through the nine points of inflexion of a given cubic has the same points of inflexion.
5. If the foci of an oval of Cassini are the extremities of a diameter of an equilateral hyperbola, prove that the two curves cut one another at right angles.
6. If the distances of a variable point from the fixed lines 0 A, OB, and from the fixed point C, are denoted by a, b, and p, and if the constant k be so chosen that the cubic p*a = kaß may have a double point; prove18t. That this point will be a cusp when 3 tan COB = tan co4; 2nd. That if D be the cusp, COA = 3 COD; and, 3rd. That a circle through C, D, and O will touch the cubic at the cusp.
7. If p1p2p3p4 are the distances of a variable point from four fixed points on a right line or a circle, show that a, b, c, d may be so determined that api + bp2 + cp3 + dp4=o for all positions of the variable point.
8. Prove that if the co-ordinates of the intersection of the line z = 0, with the polar line of any point with respect to the curve U=o be substituted in the equation of any higher polar of the same point, the result will be of the form PU+Qz2=0.
9. Show how to compute P and Q in this equation, U being a given function of x, y, and z.
10. In transforming from trilinear to three point co-ordinates, and vice versá, show that there are three points which correspond to the same three lines in each transformation, and show in what respect one of these points differs from the two others.
1. In determining a surface of the second degree, to how tions is it equivalent to be given a point and its polar plane ? To how many to be given a self-conjugate tetrahedron?
2. The perpendiculars from each vertex of a tetrahedron on the opposite face are generators of the same hyperboloid ?
3. Find the condition that it should be possible to draw to a surface of the second order-
(a). Three mutually rectangular asymptotic lines. 6). Three mutually rectangular asymptotic planes.
4. Every equilateral hyperbola which passes through three fixed points passes through a fourth. Find an analogous theorem for surfaces.
5. Find the surface generated by the intersection of two mutually rectangular planes, each of which passes through a fixed line.
6. Find the equation of the cylinder generated by a line always parallel to x =mz, y=nz, and which meets the plane of xy in the curve
22 7. Find the lengths of the axes of a section of a quadric made by a plane making given angles with the axes of the surface.
8. Find the locus of points on a surface at which the two lines coincide, which can be drawn to meet the surface in three coincident points.
9. A double point on a surface will be also a double point on its Hessian ?
10. Prove Meunier's theorem that the radius of curvature of an oblique section is the projection of the radius of curvature of a normal section.
11. The plane of two consecutive elements of a geodesic line contains the normal to the surface?
12. If A1, A, be the minors obtained from a determinant A by erasing the first line and first and second columns; B1, B2 those obtained by erasing the second line and first and second columns; C the minor ob