We shall call x' the conjugate to x and σ = o(x) the norm of x. Hence the product of any number and its conjugate in either order equals its norm. We assume that the norm of a product equals the product of the norms of the factors: = σ(x)o() σ(X), if x§ = X, (4) and shall investigate the resulting types of linear algebras. We assume also that each c; ‡ 0 in (3). (k> 1) Since this transformation is the identity X = x if = 1, we obtain an infinitesimal transformation by taking 1 = = = δί, ξι 0(i+1, j): Vijjxi For these 's, o() is unity to within an infinitesimal of the second order. Hence the increment to σ(x) must vanish identically, so that Xičici, Xičk + xrši + Σ XičjVijk (k>1), (9) where, in the final sum, i and j range over distinct values from 2, .., n, excluding k. This final sum is, therefore, absent if n=3; whence σ(X) has the term 2x22C2.X33C3 which does not occur in o(x)o(§). But C2C3 0 by hypothesis. Hence n>3. Hitherto we have not examined the conditions which follow from the final equations (3); these are (10) (11) 6. Taking n = 4 and applying (10), we see that (9) become (X1=X11+ C2X2§2 + €3X3§3 + €4X4§4, X2=X1§2 + X2§1 + Y342(X3§4— X43), X3=X13+ X3§1 + Y243(X2§4— X4§1), X4=X1§4 + X4§: + √234(X2§3 — X3§2). These transformations do not in general form a group and hence are not generated by the corresponding infinitesimal transformations employed above. Hence it remains to require that (X) = o(x)o(§) under the transformations (11). The conditions are seen to be - C3C4= — C2 Y 342, = = —C3 Y 243, C2C3 -CAY 234, C4Y234 = C2Y342 = -C3Y243, the first two of which reduce to the third by means of the last three equations. To these last can be reduced all the conditions (8) by means of (10). Applying the transformation of variables which multiplies x4, §4, X4 by Y234, and leaves the remaining x,,,, X, unaltered, we get SX1=X1§1 + €2X2§2 + C3X3§3 −C2C3X4§4, X2=X1§2 + X2§1−C3X3§4 + C3X4§3, |X3=X1&3 + X3§1 + €2X2§4−C2X452, X1=X11 + X4E1 + X2E3-X32. (111) = These are the values obtained by Lagrange in his generalization σ (x)σ(§) (X) of Euler's formula for the product of two sums of four squares. Then x = X gives the following multiplication table for the units: Se = c2, e; = c3, e = = C2C3, C2€3 = €49 €3l2 = -C4, | €24 = C2C3, €4€2 —C2¤3, €3Є4 = -C3C2, €43 = C3l2. (12) This algebra is associative and is the direct generalization of quaternions to a general field F which the writer obtained elsewhere from assumptions including associativity. The four-rowed determinants of the general number x of this algebra equals o2(x). The case c2 = C3 = -1 gives the algebra of quaternions, for which it is customary to write i, j, k instead of our units e2, 3, €4. 7. It is not very laborious to show by the above method that the cases n = 5 and n = 6 are excluded. However, Hurwitz has proved that a relation of the form σ(x)σ(§) = σ(X) is impossible if n + 1, 2, 4, 8. A slight simplification of his proof, together with an account of the history of this problem, has been given by the writer." Hurwitz made no attempt to find all solutions when n = 4. We proceed to treat this problem. Consider the case c; = -1 to which the general case may be reduced by an irrational transformation. Then σ(x) Ex. We investigate Σχ. the linear algebras having property (4), i. e., = (x2 + .. +x) (§ + ... +§) = X; + ... + X2, (13) The matrix M of this substitution has the element =~; in the kth row and jth column. If this substitution is applied to a quadratic form in X1,...,X, of matrix Q, it is a standard theorem that we obtain a quadratic form in 1,..., n, whose matrix is M'QM, where M' is the transposed of M, being obtained from M by the interchange of its rows and columns. In our problem, Q is the identity matrix I whose elements are all zero except the diagonal elements which are 1. Hence, by (13), When a homogeneous polynomial σ(x1, ..., x) of any degree has the property (4) of possessing a theorem of multiplication, the writers has proved that we may apply a linear transformation on x1, .., Xn which leave σ(x) unaltered and one on §1, ..., En which leaves o(§) unaltered such that the new algebra has the principal unit e1, so that Yık and Yjlk are both 0 if j ‡ k, and both unity if j = k. Hence M = x1M1 + +xnMn, where Yijk is the element in the kth row and jth column of M,, whence M; = I. M-M,, MM; = 1, M';M; + M;M; = = 0 Thus (15) gives (i>1, j>1,j ‡ i). (16) = 4, -M1, we have, when n The final condition (16) states that MM; is skew-symmetric. The products M2M3, M1⁄2M4, M3M4 of matrices (17) are seen at once to be = = skew-symmetric if and only if d = −y, e = y, and then MM3 = √M4, M2M1YM3, MзMA YM2. Writing i, j, k for M2, M3, YM4, we have the multiplication table of quaternions. Or we may form the matrix M and write X for the sum of the products of the elements of its kth row by 1,..., §, and take y = 1 (by multiplying x4, 4, X4 by y); we obtain (11') for C2 C3 = -1. = -1. Hence we have again obtained the quaternion algebra without assuming the associative law. The case n 8 is being investigated in this way by one of my students. 4, 1 Frobenius, Jour. für Math., 84, 1878 (59). = 2 Dickson, Linear Algebras, "Cambridge Tracts in Mathematics and Mathematical Physics," No. 16, 1914 (10-12). 3 Dickson, Bull. Amer. Math. Soc., 22, 1915 (53–61). 5 Lagrange, Nouv. Mém. Acad. Roy. Sc. de Berlin, année 1770, Berlin, 1772 (123–133); Oeuvres de Lagrange, 3, 1869 (189). Reproduced in Dickson's History of the Theory of Numbers, II, 1920 (279-281). 6 Dickson, Trans. Amer. Math. Soc., 13, 1912 (65). 7 Dickson, Annals of Math., 20, 1919 (155-171, 297). 8 Dickson, Comptes Rendus du Congrès Internat. Math., Strasbourg, 1920 (131–146) . NOVOCAINE AS A SUBSTITUTE FOR CURARE1 HARVARD UNIVERSITY Communicated by G. H. Parker, March 3, 1921 Since the recent war, the need of a substitute for the Indian arrow poison, curare, has been keenly felt in many physiological laboratories. While investigating the activity of certain local anesthetics, it was found that novocaine, in its effect upon the neuro-muscular mechanism of frogs, duplicates in many particulars the unique action of curare. If the sciatic nerve of a sciatic-gastrocnemius preparation is bathed in a strong solution of novocaine (2.5 per cent in water or in physiological salt solution) for as long as twenty minutes, no decrease in its conductivity can be observed. However, if the muscle itself is bathed in such a solution (by direct immersion or, "painting" with a camel's hair brush) the power of reacting to nervous stimulation is destroyed within three to five minutes, though ability to respond by contraction to direct electrical stimulation remains unimpaired. Thus, in the action of novocaine there is a complete duplication of the properties originally described by Claude Bernard for curare. Whether novocaine acts directly upon the end-plates of the motor fibers or upon some membrane intermediate between the plates and the 1 Contributions from the Zoological Laboratory of the Museum of Comparative Zoology at Harvard College. No. 330. muscle has not been determined. A dye that I have made by linking novocaine with a benzene nucleus was found to be physiologically active, like the 2.5% novocaine, and appears to stain the elements acted upon by the novocaine. When the stain is diazotized directly into the living muscle, by putting the tissue first into novocaine and then into a solution of the staining base, only the muscle nuclei take the stain deeply, the nervous elements of the end-plates as well as the motor fibers remaining uncolored. It seems reasonably certain, therefore, that novocaine acts upon some constituent of the neuro-muscular mechanism beyond the end-plates. The significance of the affinity of the dye for the muscle nuclei is as yet unknown. The object of this note is to direct the attention of physiologists to a convenient substitute for curare. NOTE ON A METHOD OF DETERMINING THE DISTRIBUTION OF PORE SIZES IN A POROUS MATERIAL BY EDWARD W. WASHBURN DEPARTMENT OF CERAMIC ENGINEERING, UNIVERSITY OF ILLINOIS -2 y cos 0 γο The pressure required to force mercury into a capillary pore of radius, r, is where y is the surface tension and the angle of contact. Upon this relation can be based a method for determining the effective pore diameters in a porous material such as charcoal. If pores of various diameters are present, one may determine also the fraction of the total porosity which is due to pores having effective diameters lying between any two stated limits. The procedure would be as follows: The coarsely granular sample of the thoroughly outgassed material is weighed and placed in a steel pressure bomb which is then evacuated until all adsorbed gases are removed. Pure mercury is then admitted to fill the bomb and a series of pressure and volume measurements are made at various pressures up to the highest pressure it is desired to employ. The decrease in volume, AV, accompanying a small pressure increase of Ap, in any part of the range must evidently be due to the filling of pores whose effective radii lie between the limits r and r Ar, or A blank experiment without the porous material should of course be made in order to correct for the compressibility of the mercury and the expansion of the bomb under pressure. For accurate results the compressibility of the porous material, and the variation of y and with p should also be known. |