of Edinburgh, Session 1881-82. 
353 
Then we call y the vector Ya/3 . The equation (5) from the 
quoted memoir will then he 
Ykcr + y = 0 (a) 
with the conditions 
S hr) — 0 , ( b ) 
<pk — 0 . (c) 
By the definition of </>p, 
(f>p = SaS /3p , 
and by that of y, we have 
(<!>-<b')p = Y.yp . ( d ) 
In the particular case p — k this gives, by ( b ) and (c) 
<f>k = ky ’ ( e ) 
so that, it is to be remarked, the system y, </>7q and k form a tri- 
rectangular system, variable it is true ; and Tfk = Ty. 
Treating ( a ) by S( )k, and remembering k 2 = — 1, we get 
<x = cfik — kSkcr . (f) 
Now by our equation (15), in which we introduce 
X = i, p —j , VA pi — k, 
we get 
<b' 2 k - m 2 <b'k + mf - Ycfii<f>j = 0 . 
Eliminating (fk from this and in 2 cr = mf> k - m 2 /rSA*o-, we get 
m 2 or = (f)' 2 k + (m 1 - mf>k(r)k - Y (j)i<f>j . ( g ) 
Moreover we have by ( d ) : 
Hence 
This gives : 
where we put 
(<jf> — (b')(f>k — Yy<j>k 
= kTyTcf>'k 
« - ky 2 . 
<b' 2 k = <b<fik + ky 2 . 
(7i)w 2 (r = (b<b'k — xk — Y <f>i<f>j •> 
x = - m 1 + m 2 S&cr - y 2 . 
Now this equation ( li ) represents, with changed sign, the equation 
(8) in question. Namely, if we apply the function <f> to 
p = - i$ip —jyijp - kSkp , 
