X 

 X 



its Analogues, and the Calculus of Imaginaries. 283 



would be \/irv=. 

 dw.e- w \<f>{x + 2iw\/(i R .bt), y + 2jw\^(i R .bt), z ^kws/^bt)}, 



+ 

 dw.e- w \'*lr{% + 2iw\/(—i Jli .bt),y + 2jw\/(—i K .bt), z + 2kw\/(—i Jk .bt)} 



As regards the last two results, it may be again observed that 

 they too demand interpretation, and that upon their susceptibi- 

 lity of such their practical value will depend*. 



6. It is obvious that a method of integration similar to that 

 just exhibited will apply to the equations 



©)>($)•*©)•-» 



The solutions of these equations are commonly supposed to 

 be, respectively, V x = const., 



Y 1 = const., 

 Wj = const. 

 The general solutions are however very different, and can now 

 be easily seen to be, respectively, 



{V.-^x + iy)} • {U 1 ~^~^)} = 0, 



{Y l -(j>(x + iz, y+jz)\ . {Y^^x-iz, y-jz)} = 0, 



{W l —^)(x + iw, y+jw, z + Jcw)} . {W x -"^r{oc—iw, y—jw, z—kw)}=0. 



7. In some recent speculations, Sir William Hamilton has 

 employed an imaginary unit h, distinct from i, j, k, and commu- 

 tative with them. By its aid we are enabled to arrive at some 

 interesting results. Thus the solution of the equation of the 

 vibratory motion of a flexible and slightly extensible membrane, 

 namely, ^ ; rpg ^£\ 



df ~ a \M dy*J 



* Sir William Hamilton called attention to the importance of the 

 s y mbo1 . d t . d , h d 



dx dy dz 

 in the Proceedings of the Royal Irish Academy for July 1846, as also to 

 the law of combination of this with the similar symbol 

 . d , . d h d 



l d^' +J W + *? 

 in the Philosophical Magazine for October 1847- Neither remark seems 

 to have been turned to practical account up to the present attempt to ap- 

 ply the former. The latter point seems most valuable and likely to reward 

 investigation. 



