On the Ileal Functions of Imaginary Quantities. 287 



it would therefore imply that all the rays which diverged before 

 incidence from the luminous point or primary image ^,1/', di- 

 verge after emergence from a prismatic focus or secondary 

 image, having for coordinates, 



x = x' + -yjA J/=3/': (15) 



Tfig? t y — 11 \ 



so that the last term, — (- — — f ) , of the expression (13) 



for the undulatory time V, may be considered as an aberra- 

 tional term, arising from and determining the aberration (of 

 figure, not of colour) of the prism. Accordingly Mr. Potter, 

 neglecting this aberration of figure, did not perceive this term 

 in the expression of the undulatory time, and was led to the 

 results respecting the locus of points of simultaneous arrival of 

 two near homogeneous streams, which I attempted in my last 

 paper to correct. In comparing my present notation with my 

 former, we are to make x 1 = — Z, and y f — + a ; we are also 

 to observe that the present origin of x and y is on the edge 

 of the prism. It seemed useful to give the present outline of 

 a proof of the results stated in my former paper ; because the 

 methods which I have introduced for the solution of optical 

 problems differ much from those usually received; and be- 

 cause it would perhaps be difficult, by those usual methods, to 

 investigate the influence of the prismatic aberration of figure, 

 on the undulatory time of propagation of homogeneous light. 

 Dublin Observatory, March 12, 1833. 



XL VI II. On the 'Real Functions of Imaginary Quantities. By 

 R. Murphy, Esq. M.A. Fellow ofCaius College, Cambridge*. 



T^HE following is a very remarkable property of this class 

 * of functions. 



" The real functions of imaginary quantities do not gene- 

 rally admit of maximum or minimum values." 



where a, /3, P, and Q are all real ; then it is easily seen that 



2P ■»/.(« + /3 • =!) +f(*~fi • =!) 

 and 2Q s/~\ =/(* + £ */^T) -/(*-/3 j/Cjfjj 



Moreover, if we only consider real functions, we must put 

 Q = 0, and thus establish a relation between a and (3. 



* Communicated by the Author. 



