352 
Proceedings of the Royal Society 
Then we have 
<f>e = 0 . 
Also, because of <f>‘ 'p = - g>p, we have 
cf>'e = 0 . 
t = Va'P; 
<j>p = Y( Ya'/T. p) = aS/3'p - /?'S a> . 
If we liken this with our former expression (4) we get 
= a, cq = /T, /5 = /3', ^ = - a, 
Ya i 8 = Ya 1 ^ 1 = Va'^ = e. 
We may assume 
then we get : 
a 
and we have 
Then we get by (14) 
( [g > 2 - mf> 4- m^)p = - eS pe . 
We have of course m = 0, and then 
= ~SYa/3Ya 1 j8 1 = — e 2 , 
m 2 = S(aa 1 + = ~ fi a ) — 0 • 
Also 
h> 2 p = (f > . Yep = = YeS = eS , 
because SeS must vanish in virtue of Yep = 8. Therefore our 
equation gives : 
eS - e 2 p + eS pe = 0 . 
And as S pe remains indeterminate = x, we have the solution : 
p = e~^(x + S ) , 
which is the same as the one obtained by indirect processes. 
Our second application will be to establish the equation (8) above 
quoted, namely: 
btp = x/3 + e x e 2 J3 r - -&P , 
which, when translated into another notation (to be explained 
further), will be 
- m 2 cr — xk + Y <f>i<f>j — z?k . 
We introduce, namely, for /3 — TJ/3 of Professor Tait’s Memoir, the 
letter k ; we put also 
pT/3 — pb = a . 
