154 Proceedings of the Roijal Irish Academy. 



Differentiate the first of these equations with respect to u, taking into 

 account the relations of § 10 (3), 



dv dv I — d d\L - imd, 



du dv ^' du 0iP + im<p 



((0^ + 'j/^) - imdiP) m + Mn-0) - ['ff' +^')<j, + imexP) (94' - im<t>) 



= i<pp . "^1' from (9). Hence, -p- = - i iv = « + c, (10) 



when ■It = 0, Qv = - Jfii', <Pv = J., (^ + v), ^v =Jfx{fx + „), 



.-. (§ 10 (16)) r = 20, - ^Q„ c = 2tOi - UQ,, = |tO, = - f w,, from (6a) 



.-. iv = to - ii.13 or i(«' + 2ai) = ?t + iw2. (11) 



There is a certain latitude admissible in this transformation. Thus we 

 may take any of the quantities QTRa as corresponding to W, and may 

 multiply r by any constant factor, with corresponding alterations in the 

 values of fx and v. Thus, by an imaginary transformation, taking ^ + v, 

 /x - V as the roots of the discriminant A (A), - 2iu, - 2ivi, and the 

 coordinate V as corresponding to W. 



