Hence we see that the line of vibration of the ether at any point of the wave-surface is perpendicular to the generator which passes through that point.

SECOND ADDENDUM ON THE RELATIONS OF CERTAIN SYMBOLS.

By Sir JAMES COCKLE, M.A., F.R.S.

56. AFTER forwarding the first Addendum (printed at pp. 208-210 of vol. XVII.) I received a letter (dated April 17th, 1880) from Mr. Robert Rawson, of Havant, in which he gives a proof, remarkable for its conciseness, of my Theorem II.

57. Mr. Rawson's proof is in effect as follows: Since (3) of Art. 6 is in fact equivalent to (1) of Art. 5 (see pp. 22 and 21 of vol. XVII.), consequently the first cæsura is equivalent to

d

(1)=0,

which is obviously satisfied by y1 = 1

12, St. Stephen's Road, Bayswater, W., May 24th, 1887.

ON MULTIPLE ALGEBRA.

By Prof. CAYLEY.

1. I REPRODUCE a passage from my Presidential Address, British Association, Southport, 1883. "Outside of ordinary mathematics we have some theories which must be referred to: algebraical, geometrical, logical. It is, as in many other cases, difficult to draw the line: we do in ordinary mathematics use symbols not denoting quantities, which we nevertheless combine in the way of addition and multiplication, a + b and ab, and which may be such as not to obey the commutative law ab=ba; in particular, this is or may be so in regard to symbols of operation; and it could hardly be said that any development whatever of the theory of such symbols of operation did not belong to ordinary algebra. But I do separate from ordinary algebra the system of multiple algebra or linear associative algebra developed in the valuable memoir by the late Benjamin Peirce, "Linear Associative Algebra" (1870, reprinted 1881 in the American Journal of Mathematics, vol. IV., with notes and addenda by his son, C. S. Peirce): we here consider symbols A, B, &c. which are linear functions of a determinate number of letters or units i, j, k, l, &c. with coefficients which are ordinary analytical magnitudes real or imaginary, viz. the coefficients are in general of the form x+iy, where i is the beforementioned imaginary, or √(-1) of ordinary analysis. The letters i, j, k, &c. are such that every binary combination i', ij, ji, &c. (ij not in general =ji) is equal to a linear function of the letters, but under the restriction of satisfying the associative law; viz. for each combination of three letters ij.k is =i.jk, so that there is a determinate and unique product of three or more letters; or, what is the same thing, the laws of combination of the units i, j, k, are defined by a multiplication table giving the values of i", ij, ji, &c.; the original units may be replaced by linear functions of these units, so as to give rise for the units finally adopted to a multiplication table of the most simple form; and it is very remarkable how frequently in these simplified forms we have nilpotent or idempotent symbols (i = 0 or i = i as the case may be), and symbols i, j, such that ij =ji=0; and consequently how simple are the forms of the multiplication tables which define the several systems respectively.

...

2

"I have spoken of this multiple algebra before referring to various geometrical theories of earlier date, because I consider it as the general analytical basis, and the true basis, of these theories. I do not realise to myself directly the notions of the addition or multiplication of two lines, areas, rotations, or other geometrical, kinematical, or mechanical entities; and I would formulate a general theory as follows: consider any such entity as determined by the proper number of parameters a, b, c, ... (for instance, in the case of a finite line given in magnitude and position, these might be the length, the coordinates of one end, and the direction-cosines of the line considered as drawn from this end); and represent it by or connect it with the linear function ai + bj + ck + &c., formed with these parameters as coefficients and with a given set of units i, j, k, &c. Conversely, any such linear function represents an entity of the kind in question. Two given entities are represented by two linear functions; the sum of these is a like function representing an entity of the same kind, which may be regarded as the sum of the two entities; and the product of them (taken in a determined order, when the order is material) is an entity of the same kind, which may be regarded as the product (in the same order) of the two entities. We thus establish by definition the notion of the sum of the two entities, and that of the product (in a determinate order, when the order is material) of the two entities. The value of the theory in regard to any kind of entity would of course depend on the choice of a system of units i, j, k, ... with such laws of combination as would give a geometrical or kinematical or mechanical significance to the notions of the sum and product as thus defined.

"Among the geometrical theories referred to we have a theory (that of Argand, Warren, and Peacock)) of imaginaries in plane geometry; Sir W. R. Hamilton's very valuable and important theory of quaternions; the theories developed in Grassmann's "Ausdehnungslehre," 1841 and 1862; Clifford's theory of biquaternions, and recent extensions of Grassmann's theory to non-Euclidian space by Mr. Homersham Cox. These different theories have of course been developed, not in anywise from the point of view from which I have been considering them, but from the points of view of their several authors respectively."

2. The present paper is in a great measure the development of the views contained in the foregoing extract; but, instead of establishing ab initio a linear function ai+bj+ck+... as above, I deduce this, as will be seen from the notion of addition.

X, Y,

...

3. If denote ordinary (real or imaginary) analytical magnitudes, which (as such) are susceptible of addition and multiplication, and for each of these operations are commutative and associative, then we may consider a multiple symbol (x, y, ...), any given number of letters, to fix the ideas say (x, y), susceptible of addition and multiplication according to determinate laws

(x, y) + (x', y') = (P, Q), (x, y) (x', y') = (X, Y), where P, Q, X, Y are given functions of x, y, x', y'. For greater simplicity the law of addition is taken to be (x, y) + (x', y') = (x + x', y + y'),

so that as regards addition the multiple symbols are commutative and associative. But this is or is not the case for multiplication, according to the form of the given functions X, Y; for instance, if

(x, y) (x', y') = (xx -yy', xy + yx'),

then in regard to multiplication the symbols will be commutative and associative. But if

(x, y, z) (x', y', z') = (yz' – y'z, zx' – z'x, xy' - x'y), then the symbols will be associative, but not commutative.

4. I remark here that we are in general concerned with symbols of a given multiplicity, double symbols (x, y), triple symbols (x, y, z), n-tuple symbols (x1, x2, ..., x) as the case may be, and that as well the product as the sum is a symbol ejusdem generis, and consequently of the same multiplicity, with the component symbols; this is to be assumed throughout in the absence of an express statement to the contrary. It is, moreover, proper to narrow the notion of multiplication by restricting it to the case where the terms (X, Y, ...) of the product are linear functions of the terms (x, y, ...) and (x', y', ...) of the component symbols respectively; any other form (X, Y, ...) is better designated not as a product, but as a combination (or by some other name) of the component symbols (x, y, ...) and (x', y', ...).

5. I assume, moreover, that if m be any ordinary analytical magnitude, this may be multiplied into a multiple symbol (x, y, ...), according to the law m (x, y, ...) = (тх, ту, ...).

6. As a consequence of this last assumption and of the assumed law of addition, we have for instance

(x, y, z) = (x, 0, 0) + (0, y, 0) + (0, 0, z)

=x (1, 0, 0) + y (0, 1, 0) + z (0, 0, 1);

that is, using single letters i, j, k for the multiple symbols (1, 0, 0), (0, 1, 0), (0, 0, 1) respectively, we have

(x, y, z) = xi + yj + zk,

where the letters i, j, k, thus standing for determinate multiple symbols, may be termed "extraordinaries." Each

VOL. XXII.

NN

extraordinary may be multiplied into any ordinary symbol æ, and is commutative therewith, xi=ix; moreover, each extraordinary may be multiplied into itself, or into another extraordinary, according to laws which are in fact determined by means of the assumed law of multiplication of the original multiple symbols; and, conversely, the law of multiplication of the extraordinaries determines that of the original multiple symbols; thus, if

then

(x, y) (x', y' y') = (xx' – уу', ху' – yx'),

(ix+jy) (ix' + jy')=i (xx' - yy')+j (xy'+yx')

=i'xx' + ijxy'+jiyx'+j*yy',

which expressions will agree together if, and only if, i"=i; ij=j, ji=j, j2=-i, or, as these equations may be written,

and so in general we have a multiplication table giving each square or product as a homogeneous linear function of all or any of the extraordinaries.

7. And, conversely, from this multiplication table of the extraordinaries i, j, we have

that is

(ix+jy) (ix'+jy')=i (xx' -yy')+j (xy' + yx'),

(x, y) (x', y') = (xx' – уу, ху' + yx')

the originally assumed formula of multiplication.

In the example just given we have i2=i, ij=ji=j, viz.

the symbol i comports itself like unity, and may be put = 1; we have (x, y) = x + jy, with the multiplication table