380 
“¢ Application of the above Results to the Solution of the 
Equati OP gree ey 
quation ~~ ae igs Tee 
** Writing this equation in the form 
) Sia d? 
dat * dys * az u=0, 
we can, by the properties of quaternions, reduce it to any of — 
the following forms, 
eg ee og ee ele 
de dy !dz)\de ‘dy ‘dz}" ” 
ae ae aA irae (ed 40 
etigtta) eagehe)*=% 
d aiovd d avd 
(attetig) (3.-#- ig) une. 
You employ the second of these, which leads to the solution 
5 pt 
u = oa kz) 
Si(y, 2) + ot ae) (Y, Z)s 
the symbolical form adopted by Mr. Carmichael and yourself. 
‘* Now the development of the first operating symbol will 
be obtained from (II.) by changing therein w and z to 0, y to 
Xs and zt was h r= {(=) 2% 
ea 0 e——: whence =2 | ay + Bs \ 
‘¢ We thus find 
ad d 
e(jSsk ad 
e ( o ie) cos o( = 
-2(j La us 
The development of e Cate) will be obtained from the 
above by merely changing 2 into - 2. 
