Notices respecting New Books. 365 



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 self- 



conjugate, 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 — 1), where is the angle of rotation : the second leading 



to the result that a strain $ is equivalent to a pure strain V <f (f> 



6 * 

 followed by a rotation -p^~ • An expression for the angle be- 

 V <f> (f> 



tween 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 o£ 



them well known, worked out by a new method, while here and 



there a new result is obtained. 



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 TV. E. Hamilton wrote the words : — 

 " Professor Tait, who has already published tracts on other appli- 

 cations 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 un- 

 like 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 

 certain, that the Quaternion methods enable us to solve with ease 

 many questions which are very difficult when attempted in other 

 ways. 



* The symbol denotes the operation by which any unstrained vector 

 p becomes the strained vector <f>p ; <£' is an operation conjugate to <p. 



