278 Mr. R. Carmichael on Laplace's Equation, 



It may be well to add, that the solution of such an equation as 



d*Y d 2 \ d*V 

 aW+W+-aV =Aem * C °* ny * mpZ 



is 



v= A^cos ,y.sin^ + ^ + t 



.l 2 -p 2 



3. Again, a general solution of the still higher equation of the 

 same type 



rf 2 W d*W rf 2 W d*W A , fTT , 



SF + -&- + -*?+ *F~° ' ' ' ' (IIL) 

 is given by 



Iff V A pm ] x+m. 2 y+m :i z+m 4 w 



where m v m 2 , m 3 , m 4 are connected by the relation 



tn\ + ™<i + w 3 2 + m 4 2 = 0. 

 Now, by the Calculus of Quaternions, this is equivalent to 



(m 4 -f im l +jm% + km 3 ) (m 4 — im l —jm 2 — km 3 ) = 0, 

 where t, /, and k are such operations that 



ij=—ji, jk=-kj } ki^ — ikj* 

 and the solution becomes 



+ 



To obtain the form of this corresponding to («j) and (« 2 ), we 

 assume 



rWj = rcosa, m 2 =rcos/3, w 3 =rcos7, 

 and we have 



fSAe m ' ar+,n «y +m 3* +t B • rw 



w-{ ' + 



where 



1- 



and a general solution of (III.) in the first form is 



2Ae m i'+ m # +m *'{ cos V (m, 2 + ro 2 2 + m 3 2 ) w 



4- i R . sin \/ (m^ + w? 2 2 4- w 3 2 } w 



+ 

 2Be m '* +, "2*' +w 3*{cos -/(m^ + m 2 * 4- w 3 2 )m; 



— i R . sin \/ (m, 2 -f- w 2 2 4- ?w 3 2 } w 



r 



I 



f R = i cos a +j cos # 4- ^ cos 7" 

 r 2 = 7»j 2 + m 2 2 + m 3 2 



W = 



M 



