The Kansas University Quarterly, Vol. I, No. 2, October 1892

by Various

The Great Spirit Spring Mound.

BY E. H. S. BAILEY.

The “Waconda” or Great Spirit Spring, which is situated in Mitchell County, Kansas, about two miles east of Cawker City, has been described in detail by G. E. Patrick in vol. vii, p. 22, Transactions of the Kansas Academy of Science. An analysis of the water, and of the rock forming the mound on which the spring is located, is also given.

The spring is upon a conical, limestone mound 42 feet in height, and 150 feet in diameter at the top. The pool itself is a nearly circular lake about 50 feet in diameter, 35 feet deep, and the water rises to within a few inches of the top of the basin. There is a level space on all sides of the spring so wide that a carriage can be readily driven around it.

There is but little indication of organic matter in the water of the large spring, though there is a slimy white deposit adhering to the bottom and sides, but the water is colorless, clear, and transparent. The excess of water, instead of overflowing the bank, escapes by numerous small fissures, from 10 to 20 feet down on the sides, especially on the side away from the bluff. In these lateral springs there is an abundance of green algæ, and a whitish scum, which seems to be detached from the bottom and to float to the surface. This has a slimy, granular feeling suggesting in a very marked manner hydrated silica.

The mound is situated within about 200 feet of a limestone bluff, which rises perhaps 20 feet above the level of the spring. The natural inference would be that the harder material of the mound protected it from the erosion which carried away the rock in the valley of the Solomon on the south, and the rock between the spring and the bluff.

Is it not possible however that the mound has been really made by the successive deposits from the spring? Although the mound is plainly stratified, this need not interfere with the theory, for the water may have been intermittent in its flow. The rock is very porous, and on being ground to a thin section is shown to be concretionary in structure.

An analysis of the water of the spring (loc. cit.) showed that it contained over 1120 grains of mineral matter per gallon, of which 775 grains were sodium chloride and 206 grains sodium sulphate, with 66 grains of magnesium sulphate, 41 grains of magnesium carbonate, and 31 grains of calcium carbonate. An analysis by the author shows that there are 0.874 grains of silica.

Samples of the rock composing the mound, and of the adjoining bluff were secured, and comparative analyses made, with the following results:

	COUNTRY ROCK.	GREAT SPIRIT  MOUND.
Silica and insoluble residue	2.14	4.10
Oxides of Iron and Alumina	[5] 3.22	2.66
Sulphuric Anhydride	.00	0.34
Carbon Dioxide	40.90	39.10
Calcium Oxide	51.90	49.28
Magnesium Oxide	0.63	1.15
Water and organic matter, undetermined	[6] 1.21	3.37
	100.00	100.00
Specific gravity	2.52	2.79

The rocks are entirely different in appearance and structure, that of the mound being twice as hard as that of the bluff. The former contains much organic matter as is shown by blackening when it is heated in a tube and giving off the characteristic odor. The iron is practically of the ferrous variety, probably combined with carbonic acid, and the rock contains traces of chlorides. The particular sample taken was at some distance from the spring, and had been thoroughly exposed to the weather.

The rock of the mound is of just such a character as might have been built up by deposition from the water, as it contains the least soluble constituents of the water. The process of solidification would have been assisted by the silica in the water, forming insoluble cementing silicates, as noticed by Prof. Patrick. The analysis given above shows that there is abundant silica in the water for this purpose.

Mention has been made of the organic growth in the adjacent springs. The mixed scum on being heated changes from a dull green to a vivid grass-green, and if ignited it swells up and emits an ill-smelling vapor, which is evidently nitrogenous in its character. A grayish white ash is left, which contains much carbonate of lime. This is evidently freshly deposited, as it is entangled in the algæ in granular lumps.

A specimen of the white scum, noticed above, only slightly mixed with the green algæ, was analyzed. The acid solution of the ash contains 1.26 per cent of soluble silica. This was of course a combined silica, probably calcium silicate, which becomes the cementing material in the rock. In another sample of ash, after removing all the substances soluble in hot water, the residue was found to contain 76.46 per cent of silica.

The siliceous residue from the scum was examined by Dr. S. W. Williston. It consists mostly of diatoms. He recognized

• Navicula— 2 species
• Nitzschia— 2 species
• Asteronella— 1 species.

All three genera are found both in fresh and salt or brackish water.

The green material consists essentially of Oscillaria and Confervæ. If the scum is allowed to stand for a short time a very strong sulphuretted odor is developed, strangely suggestive of salt water marshes or mud flats; and indeed the same odor is noticed in the vicinity of the spring. No characteristic salt water organisms, that should occasion this peculiar odor have, however, yet been observed here. A more extended and special study of the organic life of these interior salt water marshes and springs would be of great interest.

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On Pascal’s Limaçon
and the Cardioid.

BY H. C. RIGGS.

The inverse of a conic with respect to a focus is a curve called Pascal’s Limaçon. From the polar equation of a conic, the focus being the pole, it is evident that the polar equation of the limaçon may be written in the form:

	e		1
r =	—	cosx +	— ;
	p		p

where e and p are constants, being respectively the eccentricity and semi-latus rectum of the conic.

From the above equation it is readily seen that the curve may be traced by drawing from a fixed point O on a circle any number of chords and laying off a constant length on each of these lines, measured from the circumference of the circle. The point O is the node of the limaçon; and the fixed circle, which I shall call the base circle, is the inverse of the directrix of the conic. This is readily shown as follows:—the polar equation of the directrix is r = p / (e cosx). Hence the equation of its inverse is r = (e cosx) / p, which is the equation of the base circle of the limaçon.

If the conic which we invert be an ellipse, the point O will be an acnode on the Limaçon; if the conic be a hyperbola, the point O is a crunode. If the conic be a parabola, O is then a cusp and the inverse curve is called the Cardioid.

The limaçon may also be traced as a roulette.

Let the circle A C have a diameter just twice that of the circle A B. Then a given diameter of A C will always pass through a fixed point Q on the circle A B, (Williamson’s Diff. Cal. Art. 286) and will have its middle point on the circle A B. Now any point P on the diameter of A C will always be at a fixed distance from C and will therefore describe a limaçon of which A B will be the base circle.

The pedal of a circle with respect to any point is a limaçon. This may be inferred from the general theorem that the pedal of a curve is the inverse of its polar reciprocal, (Salmon’s H. P. C. Art. 122). For the polar reciprocal of a conic from its focus is a circle and hence its pedal is a limaçon.

The base circle is the locus of the instantaneous centre for all points on the limaçon. Let B O P be a line cutting a circle in B and Q. Let the line revolve about B, Q following the circle; the point P will trace a limaçon.

Now, for any instant, the instantaneous center will be the same whether Q be following the circle or the tangent at the point where the line cuts the circle. Therefore the instantaneous center for the point P is found by erecting a perpendicular to the line P B, through B, and a normal to the circle at Q. (Williamson’s Diff. Cal. Art. 294). The intersection (C) of these two lines is the instantaneous center for the curve at the point P. But by elementary geometry C is on the circle. Now as the line P B revolves through 360° around B, the line B C which is always perpendicular to it also makes a complete revolution and the instantaneous center C moves once round the base circle.

Below we give a list of theorems obtained by inverting the corresponding theorems respecting a conic. In these theorems any circle through the pole is called a nodal circle, any chord through the pole is called a nodal chord, and the line through the pole perpendicular to the axis of the curve is called the latus rectum. The letters e and p signify respectively the eccentricity and half the latus rectum of the inverted conic.

The locus of the point of intersection of two tangents to a parabola which cut one another at a constant angle is a hyperbola having the same focus and directrix as the original parabola.	The locus of the point of intersection of two nodal tangent circles to a cardioid which cut each other at a constant angle is a limaçon having the same double point and director circle.
The sum of the reciprocals of two focal chords of a conic at right angles to each other is constant.	The sum of any two nodal chords of a limaçon at right angles to each other is constant.
P Q is a chord of a conic which subtends a right angle at the focus. The locus of the pole of P Q and the locus enveloped by P Q are each conics whose latera recta are to that of the original conic as √2 : 1 and 1 : √2 respectively.	If P and Q be two points on a limaçon such that they intercept a right angle at the node, then the locus of the point of intersection of the two nodal circles tangent at P and Q respectively, is a limaçon whose latus rectum is to that of the original limaçon as ½√2 : 1. And the envelope of the circle described on P Q as a diameter is a limaçon, whose latus rectum is to that of the original limaçon as 1 : ½√2.
If two conics have a common focus, two of their common chords will pass through the point of intersection of their directrices.	If two limaçons have a common node, two nodal circles passing each through two points of intersection of the limaçons, will pass through the point of intersection of their base circles.
Two conics have a common focus about which one of them is turned; two of their common chords will touch conics having the fixed focus for focus.	Two limaçons have a common node about which one of them is turned; two of the nodal circles through two of their points of intersection will envelope limaçons having fixed node for node.
Two conics are described having the same focus, and the distance of this focus from the corresponding directrix of each is the same; if the conics touch one another, then twice the sine of half the angle between the transverse axes is equal to the difference of the reciprocals of the eccentricities.	If two limaçons are described having the same node and base circles of the same diameter, and if the limaçons touch each other, then twice the sine of half the angle between the axes of the limaçons is equal to the difference of the eccentricities.
If a circle of a given radius pass through the focus (S) of a given conic and cut the conic in the points A, B, C, and D; then SA. SB. SC. SD is constant.	If a circle of a given radius pass through the node (S) of a given limaçon and cut it in A, B, C, and D; then 1 —————— is constant. (SA. SB. SC. SD)
A circle passes through the focus of a conic whose latus rectum is 2l and meets the conic in four points whose distance from the focus are r₁, r₂, r₃, r₄, then 1   1   1   1   2 — + — + — + — = — . r₁   r₂   r₃  r₄  l	A circle passes through the node of a limaçon whose latus rectum is 2l, meeting the curve in four points whose distances from the node are r₁, r₂, r₃, r₄, then   r₁ + r₂ + r₃ + r₄ = 2l.
Two points P and Q are taken, one on each of two conics which have a common focus and their axes in the same direction, such that PS and QS are at right angles, S being the common focus. Then the tangents at P and Q meet on a conic the square of whose eccentricity is equal to the sum of the squares of the eccentricities of the original conics.	Two points P and Q are taken one on each of two limaçons which have a common node and their axes in the same direction, such that PS and QS are at right angles, S being the common node. Then the nodal tangent circles at P and Q intersect on a limaçon the square of whose eccentricity is equal to the sum of the squares of the eccentricities of the original limaçons.
A series of conics are described with a common latus rectum; the locus of points upon them at which the perpendicular from the focus on the tangent is equal to the semi-latus rectum is given by the equation p = -r cos 2x	If a series of limaçons are described with the same latus rectum, the locus of points upon them at which the diameter of the nodal tangent circle is equal to the semi-latus rectum, is given by the equation pr = -cos 2x
If POP₁ be a chord of a conic through a fixed point O, then will tan ½P₁SO tan ½PSO be a constant, S being the focus of the conic.	If POP₁ be a nodal circle of a limaçon passing through a fixed point O, then will tan ½ P₁SO tan ½ PSO be a constant, S being the node.
Conics are described with equal latera recta and a common focus. Also the corresponding directrices envelop a fixed confocal conic. Then these conics all touch two fixed conics, the reciprocals of whose latera recta are the sum and difference respectively of those of the variable conic and their fixed confocal, and which have the same directrix as the fixed confocal.	Limaçons are described with equal latera recta and a common node. Also the director circles envelop a fixed limaçon having a common node. Then these limaçons all touch two fixed limaçons whose latera recta are the sum and difference respectively of the reciprocals of the variable limaçon and of the fixed limaçon, and which have the same base circle as the fixed limaçon.
Every focal chord of a conic is cut harmonically by the curve, the focus, and the directrix.	Every nodal chord of a limaçon is bisected by the base circle.
The envelope of circles on the focal radii of a conic as diameters is the auxiliary circle.	The envelope of the perpendiculars at the extremities of the nodal radii of a limaçon is a circle having for the diameter the axis of the limaçon.

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Below we give a number of theorems respecting the cardioid obtained by inverting the corresponding theorems concerning the parabola.

The straight line which bisects the angle contained by two lines drawn from the same point in a parabola, the one to the focus, the other perpendicular to the directrix, is a tangent to the parabola at that point.	The nodal circle which bisects the angle between the line drawn from any point on a cardioid to the cusp and the nodal circle through the point which cuts the director circle orthogonally, is a tangent circle at that point.
The latus rectum of a parabola is equal to four times the distance from the focus to the vertex.	The latus rectum of a cardioid is equal to its length on the axis.
If a tangent to a parabola cut the axis produced, the points of contact and of intersection are equally distant from the focus.	If a nodal tangent circle cut the axis of a cardioid, the points of intersection and of tangency are equally distant from the cusp.
If a perpendicular be drawn from the focus to any tangent to a parabola, the point of intersection will be on the vertical tangent.	If a nodal circle be drawn tangent to a cardioid, the diameter of such circle passing through the cusp will be a common chord of this circle and another described on the axis of the cardioid as diameter.
The directrix of a parabola is the locus of the intersection of tangents that cut at right angles.	The base circle is the locus of the intersection of nodal circles tangent to a cardioid, which cut orthogonally.
The circle described on any focal chord of a parabola as diameter will touch the directrix.	The circle described an any nodal chord of a cardioid as diameter will be tangent to the base circle.
The locus of a point from which two normals to a parabola can be drawn making complementary angles with the axis, is a parabola.	The locus of the point through which two nodal circles, cutting a cardioid orthogonally, and making complementary angles with the axis, can be drawn is a cardioid.
Two tangents to a parabola which make equal angles with the axis and directrix respectively, but are not at right angles, meet on the latus rectum.	Two nodal circles tangent to a cardioid which make equal angles with the axis and latus rectum, respectively but do not cut orthogonally intersect on the latus rectum.
The circle which circumscribes the triangle formed by three tangents to a parabola passes through the focus.	If three nodal circles be drawn tangent to a cardioid, the three points of intersection of these three circles are on a straight line.
If the two normals drawn to a parabola from a point P make equal angles with a straight line, the focus of P is a parabola.	If the two nodal circles cutting a cardioid orthogonally and pass through the point P, make equal angles with a fixed nodal circle, the locus of P is a cardioid.
Any two parabolas which have a common focus and their axes in opposite directions intersect at right angles.	Any two cardioids which have a common cusp and their axes in opposite directions intersect at right angles.

A number of other theorems on the limaçon and cardioid are given in Professor Newson’s article in this number of the Quarterly, and these need not be repeated here.

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