ile cas ae 
63 
the second order may be made to depend on the integrations 
of two linear and conjugate equations, of which one is 
(D.-iD;z-jDy)V=0. (3) 
‘*T 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 :* 
Vige= e023 Vy (4) 
s¢ Another is 
Vayz= (Dz+tDz+jDy) E cos {2(D,? + D,?)#} Viyodz 3 (5) 
where Vy. is generally an initial biqguaternion; 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 [ 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- 
vineed 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 ijk as factors, and on which 
forms I believe that he does not now insist.—W. R. H.” 
