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of about three hundred and twenty stars is now in process of reduction, which depends on nearly ten thousand observations made since 1862. The instrument has shown a stability of mounting sufficient for an independent and accurate determination of thirty-five polar stars, while it has also proved itself capable of being used for exact right-ascensions , of time-stars by the excellence of its pivots, and its general stiffness.
The catalogue will be, if I am not mistaken, the first American Fundamental Catalogue of Right-Ascensions ; I can venture to call it so, although the still unfinished normal clock, and the lack of a good circle for solar observations, compel some reliance upon the general accuracy of other determinations. But with the requisite apparatus, about one thousand more observations would free us from even this necessity.
Mr. Oliver presented the following paper:
On some Focal Properties of Quadrics. By J. E. OLIVER.
I. Any two quadrics have one, and usually but one, common autopolar tetrahedron, T. Referred to this their equations become
U= a w+ bx2 + cya + dz? = 0,
v=aw? + B.x2 + y ya + 8z = 0, whether in tangential or in point-coördinates. Using tangential coördinates, all the quadrics
Utar= (a + a) wa +.....=0 have a common enveloping developable; and the entire system is determined by any two of its quadrics (or by any eight of its developable's planes which are not specially related]. The four quadrics that correspond to
d=-, 1=-, =-, and 1 = are plane conics in the respective planes of reference.
Deforming T and (1) together till T has one plane at infinity and the other three mutually orthogonal, and then lengthening in suitable proportions the three sets of principal axes, the rectangular pointequation of the system becomes
A+*+ B*** + 02 14 n=1, (2)
which is a homofocal system. Three of the plane curves in (1) have become the focal cunies in (2). And the fourth curve in (1) becomes in (2) the spherical circle at infinity, i. e. the intersection of any finite and finitely-distant sphere, with the plane at infinity, P. For since the cone, 2 + y + z = 0,
(3) which envelops that circle, is sensibly asymptotic to the surfaces (2) when k is indefinitely large, the common tangent-planes to those surfaces then differ not sensibly from the cone's tangent-planes, and hence envelop the circle.
Hence, as Salmon shows, a quadric's three focal curves, with the spherical curve at infinity, are the intersections of non-consecutive rays of that developable which envelops both it and the spherical curve. And they are the only intersections; for, being of the fourth degree, the developable cannot cross one of its own rays more than four times.
Any deformation that destroys a sphere, destroys with it the circle at infinity and the homofocalism of system (2), unless it replace that circle by one of (2)'s focal conics; hence the system has but four projective forms. And since, as its tangential equation shows, any four of its quadrics divide each ray of the developable in the anharmonic ratio of their k-s, this ratio must remain after deformation ; hence in the respective forms, k’ is, at the respective plane curves,
=- A’, — B”, — C?, 0%,
– B, – A2, 00?, — C'?,
co?, — C?, – B, – A”; hence the four plane conics exactly replace one another; and so do their four included groups of quadrics, since projection breaks no cyclic order.
II. The system (2) touches every point of space three times, every line twice, every plane once; except that it meets each ray of the developable throughout. A line's two planes of contact with the system are known to be mutually perpendicular, so that the boundaries of two homofocals are seen from any point to intersect at right angles if at all. For if the line touch
at (71 % 21) and (x, y2 zz) respectively, either point is on the other's polar as to each quadric; or,
is the condition that the two tangent-planes are perpendicular.
Or thus. The pairs of tangent-planes drawn from the line (21 yı 21), (19 Y2 z2), to the different surfaces, are known to form an involution, whose double planes, namely the tangents at (21 %ı 21) and (x2 Y2 z2), must form a harmonic pencil with the tangent-planes to any one surface, e. g. to the spherical circle at infinity, and hence are orthotomic.
Particular cases of this orthotomism are that of the three quadrics through one point, and the circularity of the cone mentioned in $ IV.
Now this orthotomism would preclude the existence of a common developable ; but it fails for the envelope’s rays; for since every tangent-plane from a finitely-near point to the circle at infinity has some infinite direction-cosines, while Dn, U' D no Uso cos aj · cos ag = TA2+2,2)(A2 +1:2)
X; X, are finite, [U, U., N, N., (1, . . . . Ya, being the quadrics, their normals, and the direction-angles of these,] - it must be that
Dx, U : DN, U,=0; hence (4), which is the same as
Dx, UL. Dx, U,: (cos a cos az + cos B, cos B. + cos cos y) = 0, becomes merely identical. And the other demonstration fails, because the two tangent-planes to the circle at infinity coincide.
III. According to Salmon, each ray to a spherical point at infinity is "perpendicular to itself” (whatever that may mean); which would extend the above orthotomism to the developable; a result opposite to ours in statement, but probably the same in its actual consequences. Such lines must often thus simulate self-perpendicularity, from their infinite direction-cosines having zero co-factors; and this may make
Salmon's interpretation often convenient and suggestive, though I think it is arbitrary.
If a finitely-near plane (or line) should make an angle A with itself, it would doubtless touch (or meet] a sphere's circle at infinity; for we should have
cos'a + cosa B + cos2y = cos 6, while
cosa + cosB + cosøy=1,
... some of (cosa, cos B, cos y) =.00, which is the condition of such contact; [nor need & nor o be real.] But the converse fails; for
Nor, if 8 can be > 0, need it be always 90°. Various considerations often suggest 90°, as in Salmon's beautiful instance of a circle's tangent-radius. But should not the common tangent of circles
make in like manner such angles with itself as the circles make with each other's ordinary tangents, namely,
Of course in either instance, to throw the self-contradiction, instead, upon the circle's angle e with its radius vector, we need only regard it as the limiting case of another curve or of an eccentric circle, so that e may be a function of the independent polar angle o.
IV. If quadrics U, V, W, expressed in tangential coördinates, have a common developable envelope, so have U +AY, Y +HY, W; He being a linear function of 2; for the equation
10+m V+ w=0, implies
If now y be the spherical circle at infinity, and W two separate or coincident points, we see that when U has double contact with V, or envelopes it, so does every homofocal to 'U some homofocal to V; and all the planes of contact intersect in one line or point.
U may have 2, 3, or 4 double contacts with the surfaces V tu Y, since the condition
is equivalent to three quadric conditions among (1, m, m v), which are satisfied in from 0 to 4 ways, just as three conics have from 0 to 4 common intersections. 1 of these double contacts may be replaced by 1 envelopment; or all 4 by 2 envelopments if (U, V) be cones. If U have p double contacts and q envelope-contacts with surfaces V+uY, so has V with surfaces U tay.
Of the fact that homofocals envelop homofocals, a familiar case is that each focal line of a cone U touches either co-planar focal of any enveloped quadric V; whence the circularity of that enveloping cone whose vertex is on a focal; and the consequent linear relation among the four distances of two points on one focal from two on the other, &c. Other known cases are, that U's focal lines, when not thus co-planar, are generators of some homofocal to V; and that when U is a quadric of revolution, its foci lie upon V's focals. For in neither of these three cases could the focal curve of y otherwise envelop any homofocal of v.
P. S. Feb. 10, 1866. — The above was in the printer's hands, before I was aware how much of it had been given by Salmon; but I retain it with some changes, as certain points in it may still be new.-J. E. O.
The following gentlemen were elected members of the Academy, viz. :
J. Victor Poncelet, of Paris, was elected a Foreign Honorary Member, in Class I., Section 4, in place of the late M. Struve.
Mr. Lewis M. Rutherford, of New York, was elected Associate Fellow, in Class I., Section 3.
Mr. Samuel Eliot, of Boston, was elected a Resident Fellow, in Class III., Section 3, and Mr. G. W. Hill, of Cambridge, a Resident Fellow, in Class I., Section 2.