380 
<¢ Application of the above Results to the Solution of the 
au _ eu 
. au 
Equation ant’? Hp dy asa 0 :— 
“ Writing this equation in the form 
i a § @F men 
we can, by the properties of quaternions, reduce it to any of — 
the following forms, 
@ sca ae ae ps 
de ‘dy! dz) \du ‘dy ‘dz) ”’ 
da (vd d d & Sud d 
la gohan haariage tae) 9-4 
d Gon a ad d .d 
(atta tig) (-4G-ig) ene 
You employ the second of these, which leads to the solution 
x pea us -2 Hae a. 
u=e Ow Fae) ney, z)t+e Va re) e(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 « to 0, y to 
a Cah onk dis: h rea |{(5) oy 
ay} z to #—: whence = PP IGE \ 
‘6 We thus find 
: wae: 
j—+h a ad? \t 
iba ae) = cvs { o( + 55) {4 a at 
f ae 
The development of e #(ayta) will be obtained from the 
above by merely changing a into - 2. 
