ticular (ziemlich partikulär), and is inferior in simplicity to our original introduction of the conception. On the other hand, we must not conceal from ourselves that it has a great advantage over ours. In fact, it puts immediately in evidence the half angle of rotation (w/2) required for the unambiguous description of the quaternion, while our view of Drehstreckungen deals primarily with the whole angle of rotation (w), and has then to be brought into relation with the half angle of rotation through the somewhat arbitrary rules of p. 36." The "it" (sie) of the second sentence refers presumably to Hamilton's definition of a quaternion, although grammatically it refers to their own "somewhat particular" definition immediately preceding. This definition, however, is not Hamilton's in any strict mathematical sense. What follows in the paragraph just quoted, if taken in conjunction with foregoing statements, constitutes a remarkable confession. Hamilton's definition is first criticised as being "scarcely adapted to the end aimed at," but now it is admitted to have " a great advantage" over their view of a Drehstreckung, which, we are nevertheless assured, "leads to a complete, clear, and comprehensive conception of the quaternion calculus"; and one stated reason for this great advantage is that their "complete, clear, and comprehensive conception" has to be eked out by means of certain "arbitrary rules" regarding whole angles and half angles of rotation. But, strictly and therefore mathematically speaking, their definition has to do, not with a quaternion and two vectors, but with a Drehstreckung and two Wendestreckungen, whose axes are subject to a particular limitation. A so-called quaternion Q' is represented as the quotient of two vectors v' and V; but with Q' Klein and Sommerfeld associate an angle of rotation double the magnitude of that which Hamilton would have called the angle of the quaternion v'/V. In short they use Q', Q, v and V, each and all, in a double significance. When the exigencies of analysis demand it they simply follow Hamilton and Tait-that is, their analytical work is purely quaternionic. But when there is no direct question of establishing fundamental relations among the scalar quantities involved, they endow their so-called quaternion with powers that belong, as VOL. XXIII. C Hamilton and Cayley showed long ago, to a particular quaternion Because of its peculiar form this operator, viz., operator. q ( q', involves the same four scalars which enter into the analytical expression for the quaternion'q. These four scalars have long been known to be remarkably simple functions of the half angle of rotation and of the position of the axis of rotation symbolised by the operator q ( )q. The modification introduced by Klein and Sommerfeld in their passage from the simple Drehung to the Drehstreckung is completely symbolised by the quaternion form q( ) Kq, a form already used by Tait (Proceedings, R.S.E., Vol. XIX., p. 196, 1892), while the equivalent form uq ( ) 9, where u is a scalar multiplier (in fact Klein and Sommerfeld's tensor of the Drehstreckung), was used by Tait in his earlier paper on Orthogonal Isothermal Surfaces (Transactions, R.S.E., 1873–4; Scientific Papers, Vol. I., p. 180). Thus, in their attempt to base the quaternion calculus on the conception of the Drehstreckung, the one novelty to be placed to Klein and Sommerfeld's credit is the identification of a quaternion with a very special kind of quaternion operator. Given the Hamiltonian quaternion q, it is a comparatively simple matter to pass to the required rotational operator q ( ) q1. But to pass originally from the rotation to the quaternion with which it is now known to be so intimately associated would almost certainly have proved a feat beyond the powers of any mathematical mind. For what is there in the simple conception of a rotation to suggest the presence of a quantity or operator and its reciprocal? The Examination of Sea-Water by an Optical Method. By J. J. Manley, Magdalen College Laboratory, Oxford. Communicated by Sir John Murray, K.C.B. (Read January 8, 1900). In a paper* communicated to the Royal Society, Mr R. T. Günther and the author gave an account of the results obtained from the examination of two samples of water taken from Lake Urmi, and amongst other determinations of a chemical and physical nature, were those of the refractive indices, which were performed with the aid of the Royal Society's large quartz prism and spectrometer, the latter reading by means of micrometers to 2′′ of arc. On comparing the values obtained for the refractive indices of the two samples of water with those obtained for the relative densities, it was at once apparent that the former differentiated the two samples quite as distinctly as the latter. Krummelt attempted an optical method for the examination of various samples of sea-water, by measuring their refractive indices with the aid of an Abbé refractometer. The chief objections to the use of this instrument are-(1) Its sensibility is not sufficient when the waters to be examined differ but slightly from each other in their degrees of salinity; (2) the drop of water placed upon the fixed prism must necessarily undergo a certain although small amount of evaporation before it can be covered by the second or movable prism; (3) there is a considerable degree of uncertainty as to the true temperature of the liquid contained between the prisms, even when the refractometer is supplied with a water jacket. The thermometer indicates the temperature of the water in the jacket, but, owing to the unavoidable massiveness of the prisms, and the bad conducting power of glass for heat, it is highly improbable that the temperature observed is also that of the liquid whose refractive index is being measured. *Proc. Roy. Soc., vol. 65, 1899, p. 312. The Relative Densities. In order to determine how far the optical method proposed by Krummel might be relied upon, Mr H. N. Dickson very kindly supplied the author with five samples of sea-water marked 1o, 2o, 3, 4, and 5', which differed but little from each other as regards "total salinity." The samples were first examined as follows:Using a Sprengel tube having a capacity of about 48 c.c., two series of determinations of the relative densities at 24° C. were made. The tube was first washed out with fuming nitric acid, then with distilled water, and finally with absolute alcohol; it was then dried by keeping it thoroughly heated whilst a current of air was passed through; when the tube had become quite cold, it was wiped and hung from one arm of the balance, and after an interval of five minutes its weight was determined. The tube was then charged with recently re-distilled water, and suspended centrally in a large water bath, furnished with a rocking stirrer which was kept moving by a small water motor; the temperature of the bath was indicated by a standardised thermometer reading to 0°1 C. With this apparatus the maintenance of a constant temperature, which differed very little from that of the room, was an extremely easy matter, the momentary application of a small Bunsen flame from time to time being all that was necessary. It was observed that the tube, together with its contents, assumed an almost constant temperature in about ten minutes after immersion in the bath; an approximate adjustment of the contents was then made. In every case, however, the tube was allowed to remain in the bath for twenty minutes, when the liquid was finally adjusted in the usual manner by the application of bibulous paper to the capillary. The tube was then removed from the bath, carefully wiped, again suspended from one arm of the balance, and weighed after five minutes. The contents were then discharged, the tube repeatedly washed out with portions of the sea-water to be examined, and then filled with it, and the process described above, repeated. After the first series of determinations had been completed, the tube was again thoroughly cleaned, dried and weighed, and a second series of determinations proceeded with in a manner identical with that described for the first series. The weighings were performed with a delicate long 1 beam Oertling balance and a recently standardised box of weights. Table A shows the values obtained for the different weighings in the two series. I. 16.8912 64-7332 47 8420 49 0754 49-0613 49-0561 49 0582 49.0738 II. 16 8908 64-7302 | 47-8394 49 0770 49 0607 49 0585 49.0630 49 0752 If W be the weight of a certain volume of sea-water which fills the Sprengel tube at 24° C. and w1. the weight of the same volume of distilled water, also at 24° C., then W/w, expresses the relative density at the temperature named. The values shown in Table B. were obtained in this manner. The refractive indices of the five samples of water, together with that of recently re-distilled water, were next determined with the aid of the above-mentioned large spectrometer and hollow quartz prism. Two series of measurements were made at the ordinary temperature of the room, on two different days. The bottles containing the waters to be examined were placed upon a shelf, close to the spectrometer, the day before any measurements were proceeded with; on the day of examination the water would Grams. |