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Proceedings of the Boyal Society 
We assume thus 
(7) Sy<£p = 0 and Sy^'p — O 
for any direction of p. 
The theoretical expressions (4), (5) show also that 
0(Va 1 /3 1 ) = O and <£'(Ya/3) = 0, 
namely 
(8) <f>y 1 — 0 , and <fy = 0 . 
These relations comprise the relation (7), because we have 
S ycfip = S p<f>y = Sp x (zero) = 0 , 
Syi&P ■= &pHi = 0 • 
We see thus that the properties derived from expression (4) of 
c pp , namely 
1°) m — 0 'j 7 1°) m — 0 
2°) Sy<£p=0 V l 2°) Sy^p-0 
3°) <^> yi = 0 j (3°) <£'y = 0 , 
are all consequences of eacli other, and one of the equations enu- 
merated in 1°), 2°), 3°) will have the others for consequences. 
§ 2. For the verification of the equation 
(9) "V fXy> ! [x = (<£ 2 — mfp + mfVXjx , 
in our particular case of tf>, let us first calculate the coefficients 
m l and m 2 , as they are defined in § 147 of Professor Tait’s Ele- 
mentary Treatise on Quaternions. 
If we take 
v = y, 
then, owing to <f>'y = 0, the values become 
( 10 ) 
mjSAyuy = Sycj)'X(f)'pL 
W 2 SAp,y = S (4>'X. pcy + (/>' pc. yX) . 
By the help of the theoretical expressions (4), (5) we get 
V < fXcfa'pi = V (GqSaA 1- /5 1 Sy8A)(a 1 Sa/x + /^S/Ip.) 
= Y oq/IjS . Y a/3 Y piX 
(11) Vy> , Xcfi'pL= - yfsyVXpi . 
