which is a homofocal system. Three of the plane curves in (1) have become the focal conics 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,

x2 + y2+z2 = 0,

(3)

which envelops that circle, is sensibly asymptotic to the surfaces (2) when 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 k2s, this ratio must remain after deformation; hence in the respective forms, k2 is, at the respective plane curves,

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 ( y ) and ( y ) 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 (x, y1 1), ( y ), to the different surfaces, are known to form an involution, whose double planes, namely the tangents at (x1 y1 1) 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

are finite, [U1, U2, N1, N2, a1, . . . . 72, being the quadrics, their normals, and the direction-angles of these,]-it must be that

D1 U1• DN2 U2 (cos a, cos a2+ cos B1 cos B2+ cos y cos y2) = 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 with itself, it would doubtless touch [or meet] a sphere's circle at infinity; for we should have

while

cos2 a + cos2 8+ cos2 y = cos 0,

cos2 a + cos2 3+ cos2 y = 1,

... 0 ⋅ cos2 a + 0 ⋅ cos2 8+0 cos2 y = 1 — cos e,

... some of (cos a, cos B, cos y) = ∞,

which is the condition of such contact; [nor need @ nor ∞ be real.] But the converse fails; for

does not imply

COS a∞

cos2 a + cos2 B+ cos2 y 1.

Nor, if 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

x2+y=1, found thus,

x2 + y2 = 4,

S

x cos ay sin a = = 1,
x cos ay sin a = 2,

...cos a∞,

make in like manner such angles with itself as the circles make with each other's ordinary tangents, namely,

60°, and (log 2+√3)√-1 ?

Of course in either instance, to throw the self-contradiction, instead, upon the circle's angle 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 ø.

IV. If quadrics U, V, W, expressed in tangential coördinates, have a common developable envelope, so have U+λ x, V+ μ Y, W; μ being a linear function of A; for the equation

implies

IU+m V+ W=0,

1 (U + 1 x ) + m ( V − 1 1 x ) + w = 0.

m

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+ μY, since the condition

1. U+m• V+mμ Y product

is equivalent to three quadric conditions among (l, m, m μ), 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 +μY, so has V with surfaces U+λY.

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 U 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. Rutherfurd, 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.

Five hundred and fifty-eighth Meeting.

November 14, 1865. - MONTHLY MEETING.

The PRESIDENT in the chair.

The President called the attention of the Academy to the recent decease of the eminent lexicographer Dr. Joseph E. Worcester, of the Resident Fellows.

Professor Lovering, from the Rumford Committee, delivered to the President the Rumford Medal, which had been prepared in accordance with a vote of the Academy to be presented to Professor Treadwell.

The President prefaced the presentation of this medal with the following remarks: —

At the Anniversary Meeting last May, upon the unanimous recommendation of our Rumford Committee, the medal founded by Count Rumford was by the Academy awarded to Professor Daniel Treadwell for certain improvements in the management of heat. This medal is now before us. It is the first which the Academy has ever bestowed upon one of its immediate members.

As your organ upon this occasion, before we place this testimonial in the hands of our distinguished associate, it is proper that I should briefly specify the grounds upon which your Committee proposed, and you made, this award. It is well understood, and the terms of the vote distinctly show, that this medal was awarded for an invention or an improvement in the management of heat. It is also well known that this particular improvement is a part the initial part, indeed of a series of inventions, applicable to other uses, no doubt, but through which the character of ordnance has been changed, and its power immensely increased. This was the end and aim of the improvement for which the medal is given.

We may, therefore, and we must upon this occasion, speak of this particular improvement in the management of heat in connection with the mechanical inventions which accompanied and followed it, and to which indeed the former is incidental. For the whole important series of mechanical inventions which I am to recapitulate, the Academy must regret that it has no honors which it can bestow. But their history is upon our records, embodied in the communications addressed to