280 Proceedings of Royal Society of Edinburgh. [sess. 
Lastly, by taking all of the first derived set and using the first 
part of theorem (0), there is obtained a reverse set of nine, — 
G^a 
= pa 
G 2 a 
= fa 
G 2 a" 
= qa 
G' 2 b 
= pb 
G'V 
== fb 
G' 2 b" 
-O 
"cm 
II 
G " 2 c 
= pc 
G"V : 
= f c 
G"V 
= qc 
+ fa + fa !' , 
4- pa + qa " , 
+■ qa + p"a !' , 
+ fb' + fb " , 
+ pb' + qb " , 
+ qb' + p"b " , 
+ f'c + q'c " , 
pc + qc , 
+ qc + p"c , 
and by the second part of the same theorem 
+ 
„2 
q 
2 
+ 
/2 
q 
2 
4 2 
- Gfl 
4 ,2 
+ 
G'V 
,4, ,2 
+ 
+ 
q 
/2 
+ 
q 
2 
= G a 
^4 „2 
+ 
G' b' 
+ 
+ 
q 
+ 
q 
— G a 
+ 
G' b" 
+ 
„4 2 
c , 
„4 /2 
C , 
«4 „2 
"The existence of similar results obtainable from the second derived 
set is pointed out, but separate investigation of the two sets is 
shown to be clearly unnecessary in view of the following 
theorem : — 
“ E qualibet formularum propositarum derivari posse alteram y 
si in locum quantitatum 
ABC 
A' B' C' G , G' , G" 
A" B" C" 
substituantur respective sequentes : 
B'C"-B"C' C'A" - C"A' A'B" - A"B' 
A 5 A ’ A 
B"C - BC" C"A - CA" A"B - AB" 1 1 1 
A ’ _ A ’ “ A ’ G ’ G' ’ G" ; 
BC' - B'C CA' - C'A AB' - A'B 
unde , e.g., etiam pro A ponendum — . Quod patet 
reciprocum esse, id est, ubi ilia in haec abeant, simul etiam 
haec in ilia mutariP 
