Manchester Memoirs^ F(9/. .r///. (1898), iV^. 8. 3 



Employing this convention for the present, it is easy 

 to see that if the axes are turned through an angle a, 

 retaining the origin, there is no change in j3 — 7, but 

 x-\-iy is replaced by {x-\-iy)e^'^, and 7 by 7 + a. We must 

 therefore, take the arguments of the function to be 



(^ + /» ^,t,g,pup2, , Pi-7,i^ 



Next, if we give the coordinate axes an arbitrary 

 infinitesimal displacement of translation, the analytical 

 conditions for covariancy are extremely simple, and it will 

 be sufficient to state the results as they affect the mode of 

 entrance of the several arguments \nto/(x-\-ij/). 



These are that/ij/o' ^'^'^^^ o^ty enter the function 



in the form 



and that x-\-zy will only enter in the form 



{x + iy)e-'^ - i {A /^^^ - ^) -p^ /(^^ - ^^} /sin (b\ - /3,), etc. 

 For brevity, write ^^ ^o"^ 



A.^'^^-^-^^-^/(^^-^\etc., 

 then the arguments o{f{x-\-iy^ may be taken as 



(^ + z»^"''^-/^2/sin (/3i-/32), /, g, ^3, , A-y, ft-y 



In writing them in this way an arbitrary preference has been 

 given to the boundaries of subscripts i and 2, but in any 

 case to which this method may be applied it appears 

 probable that such a choice would ultimately be made, 

 and its introduction here in no way affects the generality 

 of the results. 



This completes the conditions which the permanence 

 of form demands. 



With the form of the function limited in this way, and 

 still written /(;ir+y/), we deduce 



II - iv = e"^^ f'{x + ?V), 



