36 Mr. 0. Heaviside. On the Transformation of [Jan. 18, 



General Transformation of Wave-surface by Homogeneous Strain. 

 5. Now apply a homogeneous strain to the wave-surface. Let 



We need not suppose that the strain is pure. Use (12) in the first 

 of (10). It becomes 



0->q_- *"' ----- =0. (13) 



~ 



Now the use of vectors and linear operators produces such a concise 

 exhibition of the essentially significant properties, freed from the 

 artificial elaboration of coordinates, that a practised worker may 

 readily see his way to the following results by mere inspection of 

 equation (13), or with little more. I give, however, much of the 

 detailed work that would then be done silently, believing that the 

 spread of vector analysis is not encouraged by the quaternionist's 

 practice of leaving out too many of the steps. 



In the first place, 0~'q is the same as q0'~', if 0' is the conjugate 

 of 0. So 



in the denominator. Also, the first -1 q in (13) may be written q0'~', 

 and the postfactor 0'" 1 may then be transferred to the denominator. 

 To do this, it must be inverted, of course, and then brought in as a 

 postfactor. Similarly, the 0" 1 in the numerator may be merged in 

 the denominator by inversion first, and then bringing it in as a pre- 

 factor. We may see why this is to be done by the elementary formula 



a-'Zr'c-' = (c&a)-', (15) 



where a, 6, c are any linear operators. So (13) becomes 



= o. (16) 



Now introduce some simplifications of form. Let 



0c0' = b, 0^0' = X. (17) 



It follows from the second, and by (15), that 



tf-VV- 1 = (0/10')- 1 = X- 1 . (18) 



We also have [X] = [>] [0] a . (19) 



