PAPERS READ.

PHENOMENA OBSERVED UPON THE SOLAR SURFACE FROM 1881 TO 1884. By Rev. S. J. PERRY, Stonyhurst College Observatory, Whalley, England.

[ABSTRACT.]

THE principal subject contained in this communication was the classification of the faint evanescent markings that are always visible on the sun. The only observer who had previously made any serious study of these objects was E. L. Trouvelot whose researches at the Harvard College Observatory in 1875 were known to the author only through the few lines on the subject in Prof. Young's work on the sun. Trouvelot classed them all as "veiled solar spots," but the author of this paper has thought it more advisable to divide them into three classes.

Class 1. Ill-defined patches of a cold grayish tint, which, individually, never last for more than three minutes. They are scattered over the whole of the surface, and appear suddenly as irregular patches. The sudden appearance of many of these faint objects in the same locality and their rapid development, frequently give rise to a blurred appearance which has been so marked a feature in Dr. Janssen's magnificent solar photographs.

Class 2. Other faint spots appear at first as small, dark and fairly defined dots, and after retaining this appearance for about two minutes, suddenly spread out and cannot, at this stage of their short existence, be distinguished from spots of the first class. These are also found in every heliographic latitude and at all times, and never last more than a few minutes.

Class 3. This last class of ill-defined spots approaches much nearer to the ordinary spots than those already mentioned. They have never been seen outside the spot zones and are generally in the immediate vicinity of spot groups, but they remain indistinct throughout their whole period of visibility. They last sometimes

for a few hours but may be generally seen for two or three days. Their formation is apparently very rapid, but they disappear gradually.

This classification of the faint solar spots was followed by some remarks on the spot maximum of 1882-3; on the position of faculæ with respect to spots, offering an explanation of the apparent lagging of faculæ behind large spots; on abnormal extensions of the penumbra; on dark shadows seen in connection with faculæ ; on the colors seen in spots; and on the motion observed on several occasions when the moving body remained always projected on the solar disk from its first to its last appearance.

DESCRIPTION OF A MODEL OF A RULED CUBIC SURFACE KNOWN AS THE CYLINDROID. By Prof. ROBERT S. BALL, Royal Astronomer of Ireland, Dublin, Ireland.

[ABSTRACT.]

In a work on the "Theory of Screws" published in 18761, I discussed the various kinematical and dynamical relations of the ruled cubic surface of the third degree which is defined by the equation.

z (x2 + y2)—2mxy=0

This surface had been discovered by Plücher and he had, the writer is informed, made a model of it to illustrate its application in the theory of the Linear Complex. It does not however appear that Plücher had ever contemplated the significance of this surface except in the region of pure geometry. The dynamical interest of the surface is however very great as will be understood from the following enumeration of a few of its properties.

We first premise the following well-known theorems :

(1) Any movement of a rigid body can be produced by a twist about a screw (meaning by a twist, a rotation around an axis and a translation parallel thereto and equal to the product of the rotation by the pitch of the screw).

1 Dublin, Hodges Foster & Co.

(2) Any system of forces acting on a rigid body can be expressed by a wrench on a screw (meaning by a wrench, a force along an axis and a couple in a plane perpendicular thereto, the moment of the couple being the product of the force by the pitch of the screw).

If three twists be so related that the body by the last twist is restored to the position which it had before the first, then the three screws must lie on a cylindroid and their pitches will be given by the law,

p=c+m cos 20

Where p is the pitch of any screw on the cylindroid and where c and m are constants and is given by writing the cylindroid in the form,

Sy=x tan

2m sin 20

If three wrenches applied to a rigid body equilibrate, then the three screws must lie on a cylindroid and their pitches will be as just defined.

It follows that the composition of twists or displacements of a rigid system and the composition of wrenches follow the same laws. This is the foundation of the Theory of Screws, which will be found fully developed in the work referred to.

The model of the surface which has been figured in the frontis-. piece of the "Theory of Screws" did not exhibit the nodal line which is one of the most interesting of the features in the geometrical theory. I therefore have recently availed myself of the mechanical skill of Mr. Howard Grubb to construct a model on an improved principle. This model is now submitted to the meeting.

A brass cylinder was mounted on a dividing engine and holes were drilled into it in the calculated positions. Through each of these holes a silver wire was passed to the hole diametrically opposite. These wires intersected in the axis of the cylinder and a most beautiful and interesting model is the result. The tangent cone to the surface from any point has three cuspidal edges. These are most beautifully shown and change with every varying aspect of the model.

ON A GEOMETRICAL INTERPRETATION OF THE LINEAR BILATERAL QUATERNION EQUATION. By IRVING STRINGHAM, Professor of Mathematics, University of California, Berkeley, Cal.

[ABSTRACT.]

HAMILTON'S Solution of the equation aq + qa, e (a, a, and c being known constant quaternions) is

q = (Ka⋅c + c⋅ a1) (a22 + 2a, Sa + aKa)--1. When Sa- Sa, and Ta Ta1, this expression for p becomes indeterminate, and the equation requires a different method of treatment.

I first solve the equation

Va⋅p + p⋅ Va1 = 0, or app a1 = 0. [a⇒ Va a1 = Va ̧.] p+p We have here successively

that is

S (a+a1) Vp=0,

Vpx (aa) + y Vaa1,

p = x(a — a1) + y Vaa,+Sp,

since by the condition of the first of these equations, Vp must lie in a plane perpendicular to (a+a1). If this expression for p be substituted in the equation ap+pa1 = 0, the part involving x (aa) will vanish and we shall have remaining

(a + a1) Sp+y (a—a1) Vaa1 = 0,

from which the value of Sp is easily found. The completed expression for p results in the form

Following an interpretation first suggested to me by Professor Klein, I define, in general, every quaternion to be a directed right line in space of four dimensions. Under this interpretation, p, being the sum of arbitrary multiples of two fixed straight lines, evidently generates a plane (of the ordinary kind) passing through the origin.

In order to obtain the complete solution for a q + qa1 = c, we have only to find any value of q which will satisfy the equation, and add it as a constant to the expression above found for p. Without discussing here the methods of obtaining such a value of q-which would lead me into matters I cannot confine within the limits of this notice -I merely write down one of these values, viz.,ac, remarking that it represents geometrically the perpendicular from the origin upon the plane whose equation is aq + qa1 = c.

The complete solution of this equation appears now in the form q = ±a 1c + x (a — a1) — y a (a—1),

·

the geometrical signification of which is evident, viz., q generates a plane passing through the extremity of and perpendicular to a1c.

The relations of such planes to each other in four dimensional space, their intersections, conditions of parallelism, perpendicularity, etc., I shall discuss at greater length on another occasion.

ON THE ROTATION OF A RIGID SYSTEM IN SPACE OF FOUR DIMENSIONS. By IRVING STRINGHAM, Professor of Mathematics, University of California, Berkeley, Cal.

[ABSTRACT.]

I SHALL assume the following theorem which may be easily proved; viz.: q and q1 may be so determined that any quaternion p may be transformed into any other quaternion p1, whose tensor is the same as that of p, by means of the operator q( )qı ̄1, i. e., so that qpq=P1. The tensors of q and q1 may, without loss of generality, be assumed to be unity.

The object of this notice is to explain the geometrical significance of this operation, which may with propriety be called a rotation in four dimensional space, inasmuch as p, representing a straight line drawn from the origin to any point in the space, may, if the above theorem be true, be rotated into any other such line by means of the above operator.

If q and q1 be written in the forms

It will in other words be resolved into two constituent operators

81

=

r () r1, 8 ( ) 8, in which the special relations Sr Sr1 and Ss Ss, exist. The geometrical interpretation is made by means of these two forms.