This is the equation for Elliptic Space; that for Hyperbolic Space is of course

r being measured from the intersection of the lines, and the constants of integration determined by the condition

D= ar

when is infinitely small compared with p, which of course includes the condition D 0 when r = 0.

=

for a pair of non-intersectors, r being measured in the one case from the intersection, in the other from the points of minimum distance.

From the formulæ (3), (4), and (5) all the trigonometry of Elliptic and Hyperbolic Space can be deduced most readily. I append one or two applications, and select for my purpose important formulæ, but anything like a complete development would be out of place here. +

The differential equations (2) and (2) contain all the metrical properties of elliptic and hyperbolic space. (2) suggests that a pair of straight lines diverging at a small angle from a point might intersect again in distinct points any number of times. The proposition proved above for elliptic space generally, that all the lines radiating from any point intersect in the same second point, seems, however, to compel us to conclude that at the point where any line intersects another for the second time, it must return into itself; for a line can be brought by continuous rotation into coincidence with its prolongation, hence we must reach the same second point of intersection in whichever direction we proceed from the first point. I can see no way out of this at present; and if there is none, it would appear that we cannot get beyond double elliptic space, even if we can consistently get so far.

I may refer the reader to Frischauf, "Elemente der Absolute Geometrie," Leipzig, 1876; Lobatschewsky, Crelle, xvii. p. 295; Klein, Annalen der Mathematik, iv. p. 573, vi. p. 112, &c. ; Cayley, Annalen der Mathematik, v. p. 630.

Area of Complete Plane and Total Volume of Elliptic Space.

The area of a biangle having the infinitely small angle a is

From this result we can deduce very easily the total volume S of elliptical space (single). The locus of the most distant points on the radii through any point of space is a plane. Suppose this plane divided up into infinitely small regular quadrilaterals (squares) of side k. The volume dS contained by four radii drawn to the vertices of one of these figures is

This curious result can also be obtained by calculating the volume swept out by a complete plane rotating through 180° about any line in it.

Formula for Right-Angled Triangles.

Let ACB (fig. 18) be a triangle right angled at C. Let BAb=dA, Bb-da. CbA=B+dB. If Bm be perpendicular to Ab, then bm=dc.

Calculating the area BA in the two different ways we get

(S)

(9)

The reader will observe that these are simply the formulæ included in Napier's rules for right-angled spherical triangles. The only modification being that in hyperbolic space hyperbolic functions take the place of circular functions. In other words, the trigonometry of single elliptic space is identical with the geodetic trigonometry of a sphere, although it would not be correct to say that the planimetry of single elliptic space is identical with the geodesy of a sphere.

For hyperbolic space the analogue is the pseudo-spherical surface of Beltrami.

Parallels.

As an illustration of the application of the above formula to parallels, I shall find the parallel angle in hyperbolic space.

Taking formula IV', if we make B move off to an infinite distance, then AB becomes the parallel to CB. A is then the parallel angle corresponding to b. Now since c= ∞ we have

As an example of the trigonometry of non-intersectors, I select the following formulæ, the proof of which I leave to the reader.

If KA and LB be two non-intersectors, K and L the points of least distance, KA-LB-7, KAB-LBA-, KL-d, AB=D.

The results of (6) to (11) are given by Newcomb (Borchardt, lxxxiii. p. 293) mostly without demonstration. He assumes formula (3) as one of the axioms on which he bases his synthesis. Although I have read most of the original literature on the subject, I am more immediately indebted to Newcomb and Frischauf for the materials of the foregoing sketch.

2. Note on the Theory of the "15 Puzzle."
By Professor Tait.

The

[After this note had been laid before the Council, the new number (vol. ii. No. 4) of the "American Journal of Mathematics" reached us. In it there are exhaustive papers by Messrs Johnston and Story on the subject of this American invention. principles they give differ only in form of statement from those at which I had independently arrived. I have, therefore, cut down my paper to the smallest dimensions consistent with intelligibility.-P. G. T.]

(This

The essential feature of this puzzle is that the circulation of the pieces is necessarily in rectangular channels. Whether these form four-sided figures, or have any greater (even) number of sides, the number of squares in the channel itself is always even. is the same thing as saying that a rook's re-entrant path always contains an even number of squares. This follows immediately from the fact that a rook always passes through black and white squares alternately. The same thing is true of a bishop's re-entering

path, for it is a rook's upon a new chess-board formed by the alternate diagonals of the squares on the original board.) That there may be circulation in the channel, one of its squares must be the blank one.

Hence an odd number of pieces lies along the channel, and, therefore, when they are anyhow displaced along it, so that the blank square finally remains unchanged, the number of interchanges is essentially even.

Thus to test whether any given arrangement can be solved, all we need know is how many interchanges of two pieces will reduce it to the normal one. If this number be even, the solution is possible. To find the number of interchanges, we have only to write in pairs the numbers occupying the same square in each arrangement, and divide them into groups, such as a b c d which form closed cycles. Here there are four pairs cda'

b

in the group, which correspond to three interchanges, because

Dr Crum Brown suggests the term Aryan for the normal arrangement, with the corresponding term Semitic for its perversion. Similarly Chinese would signify the Aryan rotated right-handedly through a quadrant, and Mongol Semitic rotated left-handedly through a quadrant.

Now it is easily seen that Aryan is changed into Semitic, and Chinese into Mongol, or vice versa, by an odd number of interchanges. Similarly Aryan and Mongol, and Semitic and Chinese, differ by an even number of interchanges.

Hence any given arrangement must be either Aryan or Semitic. The former can be changed into Mongol, the latter into Chinese. Unless the 6 and 9 be carefully distinguished from one another every case is solvable, for if it be Semitic the mere turning these figures upside down effects one interchange and makes it Aryan.

The principle above stated is, of course, easily applicable to the conceivable, but scarcely realisable, case of a rectangular arrangement of equal cubes with one vacant space.