526 Proceedings of the Royal Society of Edinburgh. [Sess. 
expressions, and desire to know whether without serious alteration they 
apply in the present subject. The mode of translation is generally by no 
means obvious. We will therefore give a rule which serves in many cases. 
(1) For i, j, k write vsc v vsl 2 , vsi z where vs = — qt 2 t 3 ~ h ic i L v (2) For 
S, V, K write S, S 2 , K. (3) For any vector a write either vSa where a is a 
fictor or a where a is a second order multenion. (4) Remember that in 
the third order multiplex when vs (which otherwise behaves like a scalar) 
crosses S c we must change S c to S 3 _ c . (5) Any formula so obtained is true 
in our present subject so long as all the symbols refer to a third order 
multiplex. Sometimes the theorem so obtained can be generalised so as 
to apply to a multiplex of any order. [The reason for taking vs and i,j, k 
as above is to ensure that i 2 =j 2 = k 2 = — 1, ij — k, etc., whether d = + 1 or — 1.] 
As an example take the first of equations (7), § 4. The quaternion 
analogue is 
V a /3y = aS /3y — /3Sya + ySa /3. 
Hence in our present subject (since a, /3, y always belong to a third 
order fictorplex) we have generally 
S 2 £Ta£T fifty = CTaSCT fifty - C7/3S67yG7a + ZZy$TZaZZj3. 
Passing each vs to the left and cancelling vs 3 , 
Si<qSy = aS /3y — /3Sya -1- yS a/3. 
Similarly, the second of (7), § 4, may be proved. As another example, 
from 
V/3ySap + VyaS ftp + Va/5Syp = pSa/3y 
we at once get for a third order multiplex 
S 2 f3ySap + S 2 yaS /3p + S 2 a^Sy p = pS s a/3y = S 3 a /3y.p. 
To generalise this, let the fictorplex a/3y be that of rp 2 t 3 , but let p also 
contain t 4 , l 5 . . . . The t 4 , o 5 . . . . parts do not occur on the left, and are cut 
out from the right by operating by S 2 . Hence generally 
S 2 /3ySa p + S 2 yaS fip + S 2 a^Syp = S 2 (pS 3 a^y) J S 2 (S 3 a^y.p) . . (13) 
The following generalisation of (13) can be proved from § 6, 
^w-i a 2 a 3 .... a n Sa x p - S n _ 1 a 1 a 3 .... a„Sa 2 p + . . . . = ^ n -i(p^>n a i ^2 ■ • • • a «) • (H) 
whether p be wholly, in part, or not at all within the fictorplex cqa 2 . . . a u . 
This theorem is wanted in the generalisation to any number of integrations 
of the well-known line-surface and surface-volume integrals of quaternions 
[§ 10 below]. 
A still more general form than (14) is (in the notation of next section) 
^a(Ja(^ c = S n _^rf) (15) 
