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 se. 
“ 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. >— 
a 1)+f(w-ry-1)} 
ae pep BET lh=f(w-ry-1)} (e+ jy + kz), 
might be obtained by a process simpler, though less interest- 
ing, than that adopted by Dr. Boole. 
Putting 
2 = 7 COS ay y =r cos B, Z=7 COs y, Cos’a + Cos’ + Cos*y = 1; 
and denoting 7 cos a +,j cos (3 + # cos y by 1; the problem is to 
develop f (w +r) in the form W+iX+jY+kZ. 
Now, as w is commutative with r and 1, we may employ 
Taylor’s theorem in the present instance, and thus find 
f (wri) =f (w) +f" (w) re 5 f" (w) 19 &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(w+r)=4{(f(wtry -1l)+fw-ry-l)} 
ai (fw+ry -1)-f(w-ry-1)}, 
