1889-90.] Rev. Mr Wilkinson on Theorems in Quaternions. 155 
So that 
Y(2r+1, 2« + l) = 2C(r,*)Y.a 1 a a a?;. . . . (15) 
So also, if 
B(2 v +1, 2s+ 1) = B 4 + B 2 1" R3 "t B 4 , 
Bi + B 2 = Y. a 1 {S'(2r, 2s + 1) + B'(2 r, 2s + 1)} = C(s - 1, r^Sa^ ; 
Bj = Y. a 1 a 2 {S ,, (2r - 1, 2s + 1) + B"(2r - 1, 2s + 1)} = 0 ; 
Bo = — C(s — 1, r)a 2 Soj_a? ; 
B 4 = Y . • • • ®’2r+3^®l®2®'2r+4 • • • • • • 
= B"(2r + 1, 2s - lJSc^ag + Y.a 3 a 4 . . . a 2j . +3 S£a 2r+4 . . . - . . ; 
B^(2r + 1, 2s) = Y. £a 3 a 4 . . .a 2r+2 Sa 2r+3 . Y. a 3 a 4 . . .a 2r+3 S£a 2H _ 4 . 
B 4 = - C(r, s)V£x + Y. £{S"(2r, 2s) + B"(2r, 2s)} 
= - C(r, s) Y£x + C(r, s)£Sx + C (r - 1, s)Y. fY® , 
= - C(r, s - 1)Y. £Vx ; 
whence, since Y. £Vx = a 1 Sa 2 a; - , 
B(2r+ 1, 2s+ 1) = 0 . ...... (16) 
Thus far we have shown that, if formulae (1) to (4) are true for all 
number of vectors up to an odd number inclusive, they hold when 
the number of vectors is the next even number. 
Proceeding to the next odd number of vectors, we have 
S(2r + 1, 2s + 2) = S. a 4 B'(2r, 2s + 2) + S. a 1 B'(2s + 1, 2r + 1) , 
the sign of the second term being + . Hence, 
S(2r+l,2s + 2) = C(r-l,s+l)Sa 1 a 2 aj; . . (17) 
So too, S(2r + 2,2s+l)«C(r+l,s-l)So 1 <t 2 a?; . . (18) 
Again, Z(2r + 1, 2s + 2) = Sa 4 Y'(2r, 2s + 2) + 
+ Sa 2s+3 .... a 2r+2s+3 Y a 4 a 2 . . . a 2s+2 — . . . 
= C(r, s)Sa 4 a 2 # 4- Sa 4 a 2 . . . ct 2s _|_ 2 Ya 2s _|_ 3 ... — ... 
= C(r, s)Saja 2 a; + Sa 4 Y'(2s + 1, 2r + 1) 
= 3C(r, s)S ai a 2 a;; (19) 
So too, Z(2r + 2, 2s + 1) = 3C(r, s)Sa 4 a 2 ^ ; . . . . (20) 
And, proceeding on the same lines as before, 
if Y(2r + 1, 2s + 2) = Y 4 + Y 2 + Y 3 + Y 4 , 
