Sir Witt1am Rowan Hamivron’s Researches respecting Quaternions. 215 
VJ kG 14d) = — iq; (85) 
wk=—1; (71, 4) 
that is, 
so that the first coordinate derivative, of the second coordinate derivative, of the 
third coordinate derivative of any ordinal quaternion, is equal to that quaternion 
with its sign changed ; and all the parts of the compound assertion (72), or (4), 
are justified. 
8. We see, at the same time, by (84), that 
gk ='t3 (86) 
or that a derivation in the third mode, followed by a derivation in the second 
mode, is equivalent to a derivation in the first mode. If, on the contrary, we had 
effected the two successive derivations in the opposite order, operating first in the 
second mode, and afterwards in the third mode, we should have obtained an 
opposite result, that is, a result which might be formed from the previous result 
by changing the sign of the final ordinal quaternion: for if we operate on the 
expression (80) by &, we get 
kjq = (b, —a, d, — ce) = —7q, (87) 
giving the symbolic equation, 
k= =e (88) 
of which the contrast to the equation (86) is highly worthy of attention. Ano- 
ther contrast of the same sort presents itself, between the results of operating on 
the expression (80) by the characteristic 7, and on the expression (76) by the 
characteristic 7; for these two processes give, 
yq = (—d, —c,b, a) hq; } (89) 
gig = (d,c, — b, —a) = — ka; i 
or, more concisely, 
ok; g¢0= —k. (90) 
And, finally, we find, in like manner, by operating on (76) by &, and on (82) 
by 7, the two contrasted results, 
kiq = (—¢,d,a, — b) =Jq; 7] 
1 
tkq = (c, —d, — a, b) = — jq; j ee 
VOL. XXI. 2\¢ 
