same is true of physical quantities such as potential, temperature,
&c., throughout small regions in which their variations are
continuous; and also, without restriction of dimensions, of
moments of inertia, &c. Hence, in addition to its geometrical
applications to surfaces of the second order, the theory of quadric
functions of position is of fundamental importance in physics.
Here the symmetry points at once to the selection of the three
principal axes as the directions for i, j, k; and it would appear
at first sight as if quaternions could not simplify, though they
might improve in elegance, the solution of questions of this
kind. But it is not so. Even in Hamilton's earlier work it
was shown that all such questions were reducible to the solution
of linear equations in quaternions; and he proved that this, in
turn, depended on the determination of a certain operator,
which could be represented for purposes of calculation by a
single symbol. The method is essentially the same as that
developed, under the name of “ matrices, ” by Cayley in 1858;
but it has the peculiar advantage of the simplicity which is
the natural consequence of entire freedom from conventional
reference lines.

Sufficient has already been said to show the close Connexion between quaternions and the theory of numbers. But one most important connexion with modern physics must be pointed out. In the theory of surfaces, in hydro kinetics, heat-conduction, potentials, &c., we constantly meet with what is called d2 dz d2 .

“ Laplace's operator, ” viz. @~l-Zi?-P ag. We know that this is an invariant; i.e. it is independent of the particular directions chosen for the rectangular co-ordinate axes. Here, then, is a case specially adapted to the isotropy of the quaternion system; d d d

and Hamilton easily saw that the expression i('Zc-l-JE, -l-kg could be, like ix-l-jy+/zz, effectively expressed by a single letter. He chose for this purpose V. And we now see that the square of V is the negative of Laplace's operator; while V itself, when applied to any numerical quantity conceived as having a definite value at each point of space, gives the direction and the rate of most rapid change of that quantity. Thus, applied to a potential, it gives the direction and magnitude of the force; to a distribution of temperature in a conducting solid, it gives (when multiplied by the conductivity) the flux of heat, &c. No better testimony to the value of the quaternion method could be desired than the constant use made of its notation by mathematicians like Clifford (in his Kinematic) and by physicists like Clerk-Maxwell (in his Electricity and Magnetism). Neither of these men professed to employ the calculus itself, but they recognized fully the extraordinary clearness of insight which is gained even by merely translating the unwieldy Cartesian expressions met with in hydro kinetics and in electrodynamics into the pregnant language of quaternions. (P. G. T.)

Supplementary Considerations.-There are three fairly well marked stages of development in quaternions as a geometrical method. (1) Generation of the concept through imaginaries and development into a method applicable to Euclidean geometry. This was the work of Hamilton himself, and the above account (contributed to the oth ed. of the Eacy. Brit. by Professor P. G. Tait, who was Hamilton's pupil and after him the leading exponent of the subject) is a brief résumé of this first, and by far the most important and most difficult, of the three stages. (2) Physical applications. Tait himself may be regarded as the chief contributor to this stage. (3) Geometrical applications, different in kind from, though more or less allied to, those in connexion with which the method was originated. These last include (a) C. ]. ]oly's projective geometrical applications starting from the interpretation of the quaternion as a point-symbol# these applications may be said to require no addition to the quaternion algebra; (b) W. K. Clifford's bi quaternions and G. Combebiac's tri-quaternions, which require the addition of quasi-scalars, independent of one another and of true scalars, and analogous to true scalars. As an algebraic

1 It appears from Iolfs and Macfarlane's references that ]. B Shaw, in America, independently of joly, has interpreted the quaternion as a point-symbol.

method quaternions have from the beginning received much attention from mathematicians. An attempt has recently been made under the name of multenions to systematize this algebra. We select for description stage (3) above, as the most characteristic development of quaternions in recent years. For (3) (a) we are constrained to refer the reader to ]oly's own Manual of Quaternious (1905).

The impulse of W. K. Clifford in his paper of 1873 (“ Preliminary Sketch of Bi-Quaternions, ” Mathematical Papers, p. 181) seems to have come from Sir R. S. Ball's paper on the Theory of Screws, published in 1872. Clifford makes use of a quasi-scalar w, commutative with quaternions, and such that if p, q, &c., are quaternions, when p-I-wg=p'-l-wq', then necessarily p=p', q=q'. He considers two cases, viz. w2=I suitable for non-Euclidean space, and w2=o suitable for Euclidean space; we confine ourselves to the second, and will call the indicated bi-quaternion p-{-wg an octortion. In octonions the analogue of Hamilton's vector is localized to the extent of being confined to an indehnitely long axis parallel to itself, and is called a rotor; if p is a rotor then wp is parallel and equal to p, and, like Hamilton's vector, wp is not localized; wp is therefore called a vector, though it differs from Hamilton's vector in that the product of any two such vectors wp and wo is zero because o.>2=o . p-l-wo' where p, U are rotors (i.e. p is a rotor and wo a vector), is called a motor, and has the geometrical significance of Ball's wrench upon, or twist about, a screw. Clifford considers an octonion p-l-wg as the quotient of two motors p-l-wa, p'-l-wo'. This is the basis of a method parallel throughout to the quaternion method; in the specification of rotors and motors it is independent of the origin which for these purposes the quaternion method, pure and simple, requires. Combebiac is not content with getting rid of the origin in these limited circumstances. The fundamental geometrical conceptions are the point, line and plane. Lines and complexes thereof are sufficiently treated as rotors and motors, but points and planes cannot be so treated. He glances at Grassmann's methods, but is repelled because he is seeking a unifying principle, and he finds that Grassmann offers him not one but many principles. He arrives at the tri-quaternion as the suitable fundamental concept.

We believe that this tri-quaternion solution of the very interesting problem proposed by Combebiac is the best one. But the first thing that strikes one is that it seems unduly complicated. A point and a plane fix a line or axis, viz. that of the perpendicular from point to plane, and therefore a calculus of points and planes is ipso facto a calculus of lines also. To fix a weighted point and a weighted plane in Euclidean space we require 8 scalars, and not the 12 scalars of a tri-quaternion. We should expect some species of bi quaternion to suffice. And this is the case. Let 11, w be two quasi-scalars such that 'l']2=7], wi7=co, r;w=w2=o. Then the bi quaternion oyq-I-wr suffices. The plane is of vector magnitude éVq, its equation is 5-Spq=Sr, and its expression is the bi-quaternion nVq+wSr; the point is of scalar magnitude %Sq, and its position vector is 5, where %VBq=Vf (or what is the same, ;3= [Vr-i-q.Vr. q“1]/Sq), and its expression is r;Sq+wVr. (Note that the -if here occurring is only required to ensure harmony with tri-quaternions of which our present bi quaternions, as also octonions, are particular cases.) The point whose position vector is Vrq'1 is on the axis and may be called the centre of the bi-quaternion; it is the centre of a sphere of radius Srq'1 with reference to which the point and plane are in the proper quaternion sense polar reciprocals, that is, the position vector of the point relative to the centre is Srq'1. Vg/Sq, and that of the foot of perpendicular from centre on plane is Srq'1. Sq/Vg, the product being the (radius)', that is (Srq”1)2. The axis of the member xQ+x'Q' of the second-order complex Q, Q' (where Q=17q+wr, Q'=17q'-I-wr and x, x' are scalars) is parallel to a fixed plane and intersects a fixed transversal, viz. the line parallel to q'q'1 which intersects the axes of Q and Q'; the plane of the member contains a fixed line; the centre is on a fixed ellipse which