t 364 ] 



XL VII. Notices respecting New Books, 



An Elementary Treatise on Quaternions. By P. Gr. 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.) 

 r PHIS is the second and enlarged edition of a work first pub- 

 -*■ lished 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 Eefraction, 

 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 Quo- 

 tients of Vectors, to interpretations and transformations of Quater- 

 nion expressions, to differentiation of Quaternions, and to the solu- 

 tion 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 dis- 

 cussion, and leave the reader to draw his own inferences. Eor 

 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 b_y 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 distin- 

 guish 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. 



