

88 Prof. Tait on the Importance of 



by a single letter : — thus 



Its Cartesian form requires three equations, 



a/ = ax -f by + cz, 



y' = dx + ey+fz, 



z' =gx + hy-\-iz. 



These may be simplified, but only a little, by employing the 

 notation for a matrix. To express in quaternions the conju- 

 gate strain, a mere dash is required : thus 



while with the artificial scaffolding we must write our three 

 equations again, arranging the coefficients as below : — 



a d g 



b e h 



c f i. 



If we now ask the question, What strain will convert the 

 ellipsoid above into the unit sphere, the answer will be some 

 time in coming from the ponderous Cartesian formulae. The 

 quaternion formula assigns it at once as (/>*. 



When Gauss gave his remarkable expression for the number 

 of interlinkings of two endless curves in space, he had to print 

 it as 



1 (Y (V - as) (dy dz f - dz dy f ) + (/ -y) (dz dx' - dx dz f ) + Q' - z) (dx dy f - dy dx') 



What an immense gain in simplicity and intelligibility is 

 secured when we are enabled to write this in the form 



S.p—p 1 dpdp l m 



or as 



4*0 P J T P -tf 







so that we instantly recognize in the latter factor the vector 

 force exerted by unit current, circulating in one of the closed 

 curves, upon a unit pole placed anywhere on the other ; and 

 thus see that the whole integral represents the work required 

 to carry the pole once round its circuit. 



Without as yet defining y, I shall take, as my final 

 example, one in which it is involved. A very simple term, 



