406 Proceedings of Royal Society of Edinburgh. [sess. 
index of periodicity ( n ) as a, b, c, . . . , l, and being in number 
( n — 1), e.e., one fewer than there are units in that index. 
“The number of differing arbitrary constants thus manu- 
factured is n[n— 1). 
“ Let Ax + By + Cz + . . . + lrt = 0 be the general prime deriv- 
ative from the given equations, then we may make 
A = £PD(0 afg . . . 7c) 
B=£PD(0 bfg ... 7c) 
C=£PD(0 cfg ... 7c) 
L=£PD(0(/# . . . 7c)” (xiii. 7) 
As in the case of the other theorems, no demonstration is vouch- 
safed. In order, however, that the connection between it and 
previous work may be more readily manifest, it is desirable to in- 
dicate how it would most probably be established now. Taking 
the case where the number of unknowns is t7iree and the number of 
given equations /owr, viz. — 
a 4 x + \y + c Y z = 0 ' 
a 2 x + b 2 y + c 2 z = 0 
a 3 x + b 3 y + c 3 z = 0 
a 4 x + b 4 y + c 4 z = 0J , 
we should form an array of 4(4- 1), i.e. 12, arbitrary quantities, 
fi 9i h i 
f 2 92 ^2 
f 3 9% di 3 
A 9 4 K > 
from which we should select the multiplier |/ 2 < 7 3 /i 4 | for the first 
given equation, the multiplier \ffg 3 h^ for the second equation, and 
so on. The multiplication then being performed we should by 
addition obtain 
il* + Wf$A\y + K/s?AI 3 = 0 . 
which is what Sylvester would call “ the general prime derivative of 
