410 
Proceedings of Royal Society of Edinburgh. [sess. 
It is evident also that 
</>(p + <r) n = <f>p n -f n<j>p n ~ l cr + 3 , ^ <ftp n ~ 2 o~ 2 + . . . + <j>cr u . . (9) 
a particular case of which is 
cf>(xp) n = X n cf>p n (10) 
Prom (7) it follows that 
Str V S p</>p" — - (n + 1 )S (rcfip n 
= — (n + l)Sp<£p™ _1 o- . 
By repeated application of the oper ator - So- V we get 
( - So-^/ 1 ) r Sp 1 4>Pi n = ( ?z + l)^( w - 1 ) .... (n — r + 2)Sp<£p ,l_ V’ , (11) 
where the. suffix indicates that Vj operates on p l only. If we 
replace cr by p the expression becomes 
( - Sp V = ( n +^) n • • • • ( n ~ r + 2)Sp<£p n . . (12) 
which is Euler’s theorem for a homogeneous function of the (n + l) th 
degree. 
From (9) and (11) it follows that 
S(p + o-)cf>(p + <r) n = S pcf)p n + (n + 1)S pcf)p n ~ l cr + — ^ | ^ S p</>p n_2 (r 2 . . . 
= (1 - So-V 1 + 2 ! 1 .... )S Pl (f>p 1 
= e-ScviS p^pf (13) 
From this it follows that if F(p), a scalar function of p, can be 
expanded in a convergent series of the following form, 
n—n 
F(p) = d + Syp + 2 S pcj>p n , 
n = 1 
then (see Tait’s Quaternions , page 399), 
F(p + cr) = e -s<rV 1 Fp 1 (H) 
The general equation of a surface of the (?z + l) tlx degree can ob- 
viously be put in the following form, 
F(p) = Sp(<£ w p w + ^_ip n_1 + . . . . + <f> 1 p + y) + d = 0 • (15) 
where the t£’s are completely self-conjugate vector functions. It 
