PROFESSOR TAIT ON GREEN'S AND OTHER ALLIED THEOREMS. 79 



Then after a small interval of time t, the actual co-ordinates become 



p + to- 

 and 



p + t + t(a- — S(tV)o-) 



by the definition of V in § 17. Hence, if <p be the linear and vector function 

 representing the deformation of the group, we have 



The farther solution is rendered very simple by the fact that we may assume t 

 to be so small that its square and higher powers may be neglected. 

 If <p' be the function conjugate to <p, we have 



<p'r = t — ^VSrcr . 

 Hence 



<p T = £(<? + <p> + i(<P -<P')t 



[S (tV)o- + VSro-1 - ^V.tVVo- 



The first three terms form a self-conjugate linear and vector function of t, which 

 we may denote for a moment by tot. Hence 



$T — T7TT — p V.tVVcT , 



or, omitting f as above, 



<pT = err -„V. rarVVtr . 



Hence the deformation may be decomposed into — 1st, the pure strain zs, 2d, the 

 rotation 



Jyv„. 



Thus the vector-axis of rotation of the group is 



iVV 



2" 



O" 



If we were content to avail ourselves of the ordinary results of Cartesian 

 investigations, we might at once have reached this conclusion by noticing that 



Wcr = i & 



dzj J \dz dxj \dx dyj ' 



and remembering the formuke of Stokes and Helmholtz. 

 22. In the same way, as 



m _ _d% _chj __d% 



' dx dy dz ' 



we recognise the cubical compression of the group of points considered. It 

 would be easy to give this a more strictly quaternionic form by employing the 

 definition of § 17. But, working with quaternions, we ought to obtain all our 

 results by their help alone ; so that we proceed to prove the above result by 



