63 
the second order may be made to depend on the integrations 
of two linear and conjugate equations, of which one is 
(D,-iDz,-jDy,)V=0. (3) 
‘<I am disposed, for the sake of reference, to call this * Car- 
michael’s Equation ; and have had the pleasure of recently 
finding its integral, under a form, or rather forms, so general 
as to extend even to diquaternions. 
‘* One of those forms is the following :* 
Ves Ore POV (4) 
s¢ Another is 
Viyz = (Dz +tDz+jD. DE cos {2(D,?+ D,?)*} Vaydz; (5) 
where Vy) is generally an initial biquaternion; and where the 
single definite integral admits of being usefully put under the 
form of a double definite integral, exactly analogous to, and 
(when we proceed to Laplace’s equation) reproducing, a well 
known expression of Poisson’s, to which Mr, Carmichael has 
referred. 
*<'These specimens may serve to show to the Academy 
that I have been aiming to collect materials for future commu- 
nications to their Transactions.” 
The Secretary read a letter from Count de Mac Carthy, 
presenting several books printed at Toulon. 
* “ Note, added during printing.—Since writing the above, I have con- 
vinced myself that Mr. Carmichael had been in full possession of the expo- 
nential form of the integral, and probably also of my chief transformations 
thereof; although he seems to have chosen to put forward more prominently 
certain other forms, to which I have found objections, arising out of the 
non-commutative character of the symbols 2jf as factors, and on which 
forms I believe that he does not now insist.—W. R. H.” 
