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 (j . + k— =) (y, Z) in the 
other. Practically, then, the pee rlaannee of quaternions in 
2 
a d 
the resolution of the factor a dy se 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 g(y, 2) be an arbitrary 
function involved in the solution ofa linear equation, it is 
evident that we may satisfy the equation by replacing that 
arbitrary function by any other of the form Zi¢ (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 
Xi ie SAE! dd 
(is +J z) o2(y, 2) and (as +4 =) b2(Y> Z)s 
as to the form F i 
( is b=) i, z)3 
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- 
2 si 2 
: d ad? . f 
solying the function om ge ‘hora into two linear factors, 
