1889-90.] Rev. Mr Wilkinson on Theorems in Quaternions. 153 
where, in 
Yj, cq, a 2 , both occur in first quaternion in each term ; 
V 4 ,, 5 j last ,, ,, 
Y 2 , oq occurs in the first, and a 2 in the last quaternion ; 
M3} U 2 ,, ,, cq ,, ,, 
Then, 
Yj + Y 2 = Y. cq { Z'(2r - 1 , 2s + 2) + Y'(2r - 1, 2s + 2) } 
= 3C(r - 1, s)cqSa 2 a ? ; 
Y x = Y. cqa 2 {Z"(2r - 2, 2s + 2) + Y"(2r - 2, 2s + 2)} 
= C(f-l,s)Y.a 1 a 2 Y i r; 
Y 3 = - 3C(r — 1, s)a 2 Scq# + C (r — 1, s)Y.a 2 cL 1 Yx ; 
Y 4 = Y. oqcq . . . a 2r+ 2^ a l a 2 a 2r+d • • • a 2r+ 2s+2 — • • • 
= Y"(2r, 2s)Scqa 2 + Y.a 3 a 4 . . .a 2r+2 Y^a 2r+3 . . .a 2r+2s+2 - ■ • . 
Y^r, 2s+l) = Y^a 3 ...a 2r+1 Va 2r+ 2 ... ~ • . . + Y.a 3 a 4 . . . a 2r+2M^a 2r+3 ... 
= Y.£{Z"(2r - 1, 2s + 1) + Y #/ (2 r - 1, 2s + 1)} + 
Y. a 3 a 4 * • •a 2r+2 Y £a 2r+3 . 
Y 4 = C(r, s - l)Y*Scqa 2 + C(r, s)Y£x - 2C (r - 1, s)V.£Ya? ; 
Yj + Y 2 + Yg + Y 4 = C(r — 1, s){3cqSa 2 a; — 3a 2 Scqa? 4- Y.cqcqYd? — 2Y.£Yfl?} + 
+ C(r, s)Y£r + C(r, s - l)Y^Scqa 2 = 
= C(r - 1, s){3Y.cqa 2 Y# - 3Y:rSoqa 2 + Ya?Scqa 2 — 3Y.£Ya?} 
+ C(r, s)Y£x + C(r, s - l)Y^Soqa 2 , or 
Y(2r, 2s + 2) = C(r,s)Ycqa 2 ;r; . . (11) 
In our formulae we may, of course, write r + 1 for r and s - 1 
for s, and thus obtain in the same way, 
Again, if 
Y(2r + 2, 2s) = C(r+l,s-l)Ya 1 o 2 a?; . 
B(2r + 2, 2s) = B 1 + B 2 + B 3 + B 4 , 
Bi + B 2 = V.cq {S'(2r + 1, 2s) + B'(2r + 1, 2s)} 
= C(r — 1, s)a 1 Ba. 2 X + C(r, s)Y.cqYa 2 # ; 
B 1 = Y.a 1 a 2 {S"(2r, 2s) + B"(2r, 2s)} 
= C(r, s)£$x + C(r - 1, s) Y.fY* ; 
( 12 ) 
