217 
terpretation was absolutely necessary, I took the liberty of so- 
liciting the attention of mathematicians to this point. 
«‘ Having been honoured with communications from Eng- 
land and France in connexion with this paper, I resumed the 
subject in the early part of the year 1853, and entered into 
correspondence with Sir William Hamilton. With his valu- 
able assistance I hoped to be able to overcome two difficulties 
which seemed to lie in the way of interpretation. It appeared 
desirable that the form of solution should be rendered more 
purely symmetrical by the introduction of the third imaginary 
unit &, and that by the aid of the same new element the cha- 
racter of the solution might be rendered more purely spatial. 
In one sense this form is undoubtedly spatial. If, however, 
we extract from it the explicit vector-unit, we get 
¢cosa+j sina, 
which, as_referrmg to an unit circle is planar, whereas it 
would be desirable that the explicit vector-unit should be 
2 cos a+j cos 3 +h cos y, 
referred to the unit sphere. 
“In the month of January, 1854, Sir William Hamilton 
pointed out the necessity of introducing some modification in 
the form of the solution as stated, arising out of the non-com- 
mutative character of the terms x + iz, y+jz,and x — iz, y—jz. 
‘‘ In the early part of the present year this modification 
was supplied by Professor Graves, but the same objections lie 
against the modified form : 
V= MO (a+ iz, y+ jz) + MY (#- iz, y -jz). 
In the first place this form is not purely symmetrical ; and in 
the second place, its character is not purely spatial. For these 
reasons it seems, I would say with all due respect, improba- 
ble that any interpretation of this form can be devised which 
will meet the requirements of physical research. 
