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of one being on a line through Os perpendicular to 0102 and the diameters of the other being segments of tangents to Es cut off by the tangents at A and 03.1 It should also be noted in this connection that the point T is the pole of the line drawn from the point of tangency of any two of the circles to the center of the third circle with respect to the conic passing through that center.

Also if there is drawn at D a line perpendicular to 0102 and TC1 is produced intersecting this line in S, then N, S and O1 are collinear.

THEOREM: If three circles are mutually tangent and tangents and normals be drawn to the three associated conics at the points of contact and the centers of the three circles, and if the normals of two of the conics be chosen, these will intersect by twos on a tangent to the third conic.

Proof: Consider the lines O1N, O2N, 001, 002. These intersect in the points O and N which are on the tangent to E3.

THEOREM: The axes and normals to two of the conics, together with the tangents to these two conics drawn at the centers of the circles determine two perspective triangles whose center of perspectivity is the intersection of the axis and normal to the third conic.

Proof: Consider the lines O1N, O2N, 001, 002, O1T, O2T. These intersect by twos on the line TN. They may therefore be considered as the sides of two perspective triangles. Let the corresponding sides be

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These determine the perspective triangles A12A13A23 and B12B13B23 and the center of perspectivity is the point F on 0102 (see Fig. 4 where F is marked). But this point is also on the normal OB12.

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THEOREM: If the three axes and the three normals to the three associated

1 See Conics of Apollonius, Book III, prop. 45 (Heath's ed., p. 114).

conics be drawn the axis and normal to each conic being taken as corresponding sides, they form two perspective triangles with T as center of perspective.

THEOREM: The four axes of perspective of the three circles and the six lines, three of which are normals and the other three are the tangents to the three conics at the three centers of the three circles, determine a quadrangular-quadrilateral configuration, whose diagonal triangle is the triangle determined by the three centers of the three circles.

In connection with the above discussion it should be noted that it furnishes a method for constructing points on a conic. For let 01, 02 and 03 be any three points on a line, and let the perpendicular be drawn at 03 and let N be any point in the perpendicular. Let 01 and O2 be points such that we have the order 010203 or 020103. Then from N draw lines to 01 and 02, making the angles NO103 and NO203, and draw from 01 and 02 the lines 010 and 020 such that

NO103 = 4001N,

ZNO2O3 = 4002N.

Then the point 0 is on an ellipse. If we have the order 010302, 0 will be on a hyperbola, and if O1 or O2 is at infinity we have a parabola. And in each instance O1 and O2 are foci of the conic. This also gives a method for establishing a (1, 1) correspondence between the points of a conic and the points of a straight line.

7. Şome Projective Properties of the Figure in Section 2. Let A and C be the ends of the diameter 0102 of the circle (O2) (Fig. 1), and let there be drawn from A lines to S, and from C lines to Sn', and let C and C' be the angles that these lines make with AC. Then

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n

By means of equations (16) and (17) we may find the equations of the lines AS and CSn', a solution of which reveals the fact that the line AS, and the line C'S' intersect on a line through S1 perpendicular to AC in points whose ordinates are 2pm.1

1

By very simple analytical considerations we may prove the following

THEOREM: The triangles Sn+1SnSn-1 and Sn+1'Sn'S-1' are perspective and their center of perspective is the external center of perspective of the circles 01 and 03.

THEOREM: The locus of the point of contact of two tangent circles which are tangent to two given tangent circles (internally tangent) is a circle whose center is the center of perspective of the two given circles and whose radius is the harmonic mean between the radii of the two given circles.

1 In this connection see Steiner, p. 69 and ff.

Proof: The center of perspective, P3, of the two given circles, O1 and O2, has the same power with respect to all circles tangent to these two in a given way. Therefore, the locus of the point of contact of any two such circles which are tangent to each other is a circle orthogonal to them all.

THEOREM: The circle with P3 as center and P3A as radius cuts every Cn orthogonally.

Proof: Let there be drawn with T as center and TA as radius a circle. This will pass through C1 and C2. Therefore C1C2P3 will be the radical axis of the circle just drawn and Cn. The circle with P3 as center and P3A as radius is orthogonal to the circle with center T. It is therefore orthogonal to every Cn

SOME VANISHING AGGREGATES CONNECTED WITH CIRCULANTS. By W. H. METZLER, Syracuse University.

In the course of certain investigations on Lagrange's Equation for circulants1 by Dr. Muir2 in 1912 attention was called to the vanishing aggregate:

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where a, b, c, d, e are the elements of a circulant of order five. He obtained it as the coefficient of the first power of x in Lagrange's equations, which power (as well as all the odd powers) was proven not to exist, and next enunciates the following general theorem: If the elements of the first column of any odd-ordered circulant, axisymmetric with respect to the principal diagonal, be replaced by units, the sum of the complementary minors of the elements in the places (2, 2), (3, 3), ··, (n, n) vanishes.

He next points out that in the case n = 7 we may substitute for

[2,2]1 + [3, 3]1 + [4, 4]1 + [5, 5]1 + [6, 6]1 + [7, 7]1

=

0,

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where [p, ql, denotes the complementary minor of the element in the pth row and qth column after the rth column of the circulant has been replaced by units.

1 For the purposes of this paper the following definition will be assumed: If each row of a determinant may be derived from the preceding row by passing the first element over all the others to the last place, the determinant is called a circulant.

2 "Lagrange's determinantal equation in the case of a circulant," Messenger of Mathematics, New Series, vol. 41, March, 1912.

The object of this note is to investigate these aggregates a little more closely and from a somewhat different viewpoint, and to determine those that are fundamental.

By equating the coefficients of odd powers of x to zero we would get various vanishing aggregates connecting minors of even order but these aggregates would not be fundamental.

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which obviously vanishes since two columns may be made identical, we have

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From the properties of circulants of odd order we have

1. When 1 + r is even

(1)

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[1, 1]2 + [4, 4]2 + [5, 5]2 + [2, 2]1 + [3, 3]1 + [4, 4]1 = 0. The general law for (2) is, in the case of circulants of odd order,

(4)

(− 1)[1, r], + (− 1)-1[1, s]t + (− 1)11[1, t], = 0.

(5)

The general law for (1) is

(− 1)^1[1, r], + (− 1)o1[1, s]t + (− 1)11[1, t]u + (− 1)u−1[1, u], = 0, (6)

but, as has been seen, (6) is made up of two of (5).

The method here used will give vanishing aggregates of minors of any order. Thus from

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