XLVII. Notices respecting New Books. An Elementary Treatise on Quaternions. By P. G. TAIT, M.A., Formerly Fellow of St. Peter's College, Cambridge, Professor of Natural Philosophy in the University of Edinburgh. Clarendon Press Series. Oxford. 1873. (Pp. 296.) THIS HIS is the second and enlarged edition of a work first published in the year 1867. It is designed to render the subject of Quaternions "intelligible to any ordinary student;" but it must be understood that the " ordinary student" means one who is already familiar with such subjects as Surfaces of the Second Order, Homogeneous Strain, the Theory of Double Refraction, Electrodynamics, &c. as commonly treated, and so is likely to be interested in seeing the same subjects treated by a new method. It must be a matter of congratulation to all who are interested in Quaternions that a second edition of the present work should be called for after the lapse of only six years; and this fact probably justifies the opinion of the author, that "there seems now at last to be a reasonable hope that Hamilton's grand invention will soon find its way into the working world of science." Passing to the contents of the volume, we may say that the first five chapters are devoted to an exposition of the principles of the science, viz. to Vectors and their composition, to Products and Quotients of Vectors, to interpretations and transformations of Quaternion expressions, to differentiation of Quaternions, and to the solution of Equations of the first degree. These chapters occupy 103 pp. The remainder of the book is taken up with applications-in the first place to Geometry, in the next to Kinematics and Physics. Owing to the great variety of subjects treated, it is not easy to give a satisfactory idea of the contents; perhaps our best course will be to state with some minuteness the contents of a single discussion, and leave the reader to draw his own inferences. For this purpose we will take the articles on Homogeneous Strain, a branch of Kinematics which Professor Tait has made his own, having published three distinct accounts of the subject. The author first shows that the determination of a vector whose direction is unchanged by strain depends on the solution of a cubic equation with real coefficients, and obtains the form which this equation takes when the mass is rigid. He then shows that a mass initially spherical becomes an ellipsoid after strain, and, on the other hand, that a mass spherical after strain was ellipsoidal before strain, the axis of the ellipsoid in either case corresponding to a rectangular set of three diameters of the sphere. After defining a pure strain, he obtains a criterion by which to distinguish it from other strains, and proves that two pure strains successively applied give a strain accompanied by rotation. "The Viz. Treatise on Natural Philosophy,' vol. i. pp. 99 &c., 'Elements of Quaternions,' p. 210 &c. (the work noticed in the present article), 'Introduction to Quaternions,' p. 180 &c. simplicity of this view of the question leads us to suppose that we may easily separate the pure strain from the rotation in any case, and exhibit the corresponding functions." In fact, when the linear and vector function expressing the strain is not selfconjugate, it may be broken up into pure and rotational parts in various ways, two of which are specially noticed :-the first leading to the theorem (which does not seem to have been previously noticed) that there is always one and only one mode of resolving a strain into the geometrical composition of the separate effects of (1) a pure strain, and (2) a rotation accompanied by uniform dilatation perpendicular to its axis, the dilatation being measured by (sec 0-1), where 0 is the angle of rotation; the second leading to the result that a strain is equivalent to a pure strain followed by a rotation An expression for the angle between any two lines and planes in the altered state of the body is then found, and the locus of equally elongated lines shown to be a cone of the second order. Finally, the properties of a simple shear are demonstrated. This is perhaps a fair specimen of the contents of the volume. It contains a large number of theorems, most of them well known, worked out by a new method, while here and there a new result is obtained. * φ In fact one of the main objects of the work is to show how questions hitherto worked out in other ways can be treated by Quaternions; and particularly to do this with respect to questions of physics. It seems to have been in consequence of applications to such questions that Sir W. R. Hamilton wrote the words:"Professor Tait, who has already published tracts on other applications of Quaternions, mathematical and physical, including some on Electrodynamics, appears to the writer eminently fitted to carry on happily and usefully this new branch of mathematical science, and likely to become in it, if the expression may be allowed, one of the chief successors to its inventor." (Elements, p. 755, note.) To work out old results by the new method is an indispensable preliminary to a further advance; and that further advances will, in fact, be made by the new method seems to be the general opinion of those who have mastered it. Perhaps the case may not be unlike that which actually occurred in the progress of Physical Astronomy. Newton's successors found it necessary to work out his results by new methods before they could advance further; and to repeat his work in a new form required no small expenditure of labour and genius. To what extent the resemblance will hold good in the present case remains to be seen; but this at least is that the Quaternion methods enable us to solve with ease hich are very difficult when attempted in other ques enotes the operation by which any unstrained vector hed vector op'; p' is an operation conjugate to . bisected, and the portions of the bisecting lines A E and BD contained between the vertices A, B and the opposite sides B C and AC are equal: to prove that the angles and B are also equal. Find a point F, so that FA=AD and FE line FB. I. We have AF AD FE AB II. AB, and draw Therefore But . (1) FE AB. III. From which follows the equality of triangles FBA and FBE, considering that Capt. J. Herschel on a new form of Calendar. 357 FA=BE; therefore BE-AD, From III. follows and by construction FA-AD;J from which results the equality of triangles A B E and A BD, and therefore a=B. Quod erat demonstrandum. F. G. HESSE. XLIV. On a new form of Calendar, by which the Year, or Month, or Month-day, or Week-day may be readily found when the other three components of a date are given. By Captain J. HERSCHEL, R.E.* HE Tabular Calendar annexed is perhaps as compact in form and simple in use as is possible. Its use may be stated thus:-Of the Year, Month, Day, aud Week-day, given any three to find the fourth. If we arrange these thus, D M Y W, it is clear that M and D, Y and W will each fix one of the signs or hieroglyphs in the centre of the Table; and in order that the day of the month shall fall on a certain weekday in any given year, these signs must be the same. Thus, Apr. take the case of 13 1874 Mon. April 13, and Monday in 1874, both indicate *. We have now only to suppose one of these unknown to see that the sign indicated by the other pair will show all the possible solutions. Thus :— 1. Y unknown; M and D indicate * (or any other sign, ac- .e ut ank It is obvious that the signs are pure symbols for the occasion and are for this very reason chosen beterogeneous so as to have a meaning or order, per se. necessary to add that, where the date Ty or February of a bisextile year, the herwise, or man generally, the italic M ahle Y. Thus: by the Author. Here raging دو rt itself, on thirtyations, ard will find that 48 miles per h as 73.6 miles rapid rate is 34 |