384 
a solution which becomes interpretable by developing the cir- 
cular functions. 
<¢ Now in what respect does this solution differ from the one 
furnished by the quaternion method, (III.)? Merely in this, 
that the arbitrary function ¢, (y, 2) in the one, is replaced by 
the arbitrary quaternion function (7 7 + kh— =) (y, 2) in the 
other. Practically, then, the ae ia of quaternions in 
2 
GG @& 
the resolution of the factor ae ae apt a — merely leads to the 
substitution of an arbitrary function involving quaternion co- 
efficients for an arbitrary function of the ordinary species. 
Might we not, then, if there be any advantage in the result, 
introduce the change at once? If ¢(y,z) be an arbitrary 
function involved in the solution of a linear equation, it is 
evident that we may satisfy the equation by replacing that 
arbitrary function by any other of the form 2i¢ (y, z), 7 being 
susceptible of any system of constant values, real or imaginary. 
Tt is seen that the quaternion analysis employed from the be- 
ginning leads equally to the forms 
SR d .d 
(is +z] o2(¥y, z) and (as +4 =) 2 (Ys Z)s 
as to the form 
Pee 
iz i b=) 0 z); 
and you correctly observe that any of the separate terms 
affected by distinct imaginaries equally satisfy the equation. 
‘¢ T offer no apology for making these observations. I am 
sure that your object, like mine, is the discovery of truth alone. 
The application of quaternions to the solution of partial diffe- 
rential equations is a subject deserving of being thoroughly 
investigated; partly because of the analytical interest attach- 
ing to the inquiry, and partly because the possibility of re- 
a lia 
solving the function — + = + — into two linear factors, 
dx dy dz 
