376 
preted. The compound symbols which operate upon the 
arbitrary functions f, (y, Zz), f2(y, 2), are both functions of 
the symbol jxD, + kxD,, which is, in fact, a special form of a 
quaternion. It is what the quaternion p+ gi + 7j + sk be- 
comes if we make p = 0, g=0, r=aD,, s = xD, assumptions 
which do not violate any of the formal conditions to which 
quaternions are subject. For the symbols xD,, 2D, combine 
with each other like constant quantities. 
“It appears to me, therefore, that the operating symbols 
being functions of symbolical quaternions, the most natural 
method of seeking their interpretation is to refer them to the 
more general problem of the development of a function of a 
quaternion. This development constitutes the object of the 
fundamental theorem to which I have referred. It is contained 
in the following 
PROPOSITION. 
«To develop f (w + ix + jy + kz) in the form of a simple 
quaternion, i. e. in the form W+iX+ jY+kZ, where 
W, X, Y, Z are functions of w, x, y, 2; it being given that 
Ff (w) is capable of being developed in ascending powers of w 
by Maclaurin’s theorem. 
«¢ Since f(w) is capable of being expressed in a series of 
the form 4 + Bw + Cw? + Dw’ + &e., it is evident that if we 
represent the quaternion w + iz + jy + kz by Q, the function 
F(Q) will be capable of expansion in the corresponding series 
A+BQ+ CQ’? + DQ + &.(1), for Q combines with sym- 
bols of quantity just as if it were itself'a symbol of quantity. 
We might indeed consider f( Q) as intelligible only by means 
of its development in a series of integral powers of Q. Thus 
we might take as the very goa of e° that it represents 
the series 1 + Q+—— Q@+ &. It is only by 
Q? + 
1.2 l. ae 3 
reference to such a series that we can see how a function of a 
quaternion such as ¢° can be reduced to a simple quaternion. 
For the several powers Q’, Q?, &c. assume by actual involu- 
mT. 
