1889-90.] Rev. Mr Wilkinson on Theorems in Quaternions . 151 
C. Fundamental Cases. 
By fundamental cases we mean cases in whick either one of the 
numbers r , s, in (1), (2), (3), (4) is unity. Some of these are the 
cases considered in my former paper, which proved that, 
B(l, 2n + 1) = 0 ; 
B(l, 2 n) =Va 1 a. 2 x; 
B(2, 2 n) LvojOjja;; 
B(2, 2?z+ 1) = 0 ; 
of course we have, identically, and at once, 
S(l, r) = S(r, 1) = B(r, 1) = 0 ; 
and my former paper also showed that, 
S(2, 2 n) = (n + lJSa^x ; 
S(2, 2n + 1 ) = nSa^a^e ; 
and so in other cases, and for Vs and Zs. But as we shall establish 
our formulae (1) to (4) by an induction, it will be well to put down 
the following simple identities : — 
S(1,1) = 0; Z(1,1) = 0; Y(l, l) = 2Ya 1 a 2 ; B(1,1) = 0; 
S(l, 2) = 0; Z(l, 2) = 3Sa ia2 a 3 ; Y(l, 2) = 0; B(l, 2) = Y.a^og ; 
S(2,l) = 0; Z(2,l) = 3Sa ia2 a 3 ; Y(2, 1) = Ya 4 a 2 a 3 ; B(2,l) = 0; 
S(l, 3) = 0; Z(l, 3) = 0; Y(l, 3) = 2Ya 1 o 2 a 3 a 4 ; B(l, 3) = 0; 
S(2, 2) = 2Saja 2 a 3 a 4 ; Z(2, 2) = 0 ; Y(2, 2)=Ya 1 a 2 a 3 a 4 ; B(2, 2)=Ya 1 a 2 a 3 a 4 
S(3, 1) = 0; Z(3, 1) = 0; Y(3, 1) = 2Y ai a 2 a 3 a 4 ; B(3, 1) = 0. 
2sTo difficulty whatever presents itself in any one of these cases. 
So we leave them for the beginner. All others have, doubtless, in 
some form or other, met with them frequently before. 
So, too, we leave such formulae as, 
Z(l, 2r) = 3S(3, 2r-2); 
V(2r+1, 1) = Y(l, 2r + 1) = 2Y. a } o^c ; 
Y( 2r, 1 ) = Y. a l a 2 x . 
Z(2, 2n+ 1) = 3S(3, 2 n) ; 
Z( 2, 2n) = 0 . 
V(2, 2n) = B(2n, 2) = nVa 1 a 2 x; 
Y(2, 2n + 1) = (n + lyVa^x ; &c. &c. 
