385 
seems at first sight to promise material aid in the solution of a 
problem of peculiar physical importance. The latter consi- 
deration appears to have been present to your own mind. I 
have now stated to you the reasons which have led me to en- 
tertain a different opinion. 
«‘ Believe me to be, my dear Graves, 
«« Yours very truly, 
«¢ GEorGE Boo.e. 
«“ The Rev. Dr. Graves.” | 
Dr. Graves remarked that the general expression given by. 
Dr. Boole for the development of a function of a quaternion, 
Viz. :— 
f (w+ iat jy t hz) =3(f(w+rv-I) +f (w-ry-1)} 
4 (f(w+ry—-l=f(w-ry-1)} (e+jy + he), 
: might be obtained by a process simpler, though less interest- 
ing, than that adopted by Dr. Boole. 
Putting 
* 
@=7 COS a, Y=7 C08, Z=7 COS y, COS’a + COS’P + Cos*y = 1; 
and denoting i cos a +j cos 3 + # cos y by 1; the problem is to 
develop f (w +7c) in the form W+iX +7 Y+kZ. 
Now, as w is commutative with r and 1, we may employ 
Taylor’s theorem in the present instance, and thus find 
S(w+n)=f(w)+f' @) en (w) r?— &e. 
The symbol « being a square root of negative unity, this 
development will be precisely similar to what we should have 
obtained if we had sought that of f(w+ry/-1); save only 
that . stands in the place of /-1. Consequently, we have 
f (wt) =4(f(wtry -1)+fw-rv- 1} 
U 
tari {f(wt+ry - 1) ~ f (w-ry -1)}; 
