160 
Proceedings of Royal Society of Edinburgh. [sess. 
Theorema 2. Datis , £ 2 . . . , £ n ut functionibus ipsarum 
v 1 , v 2 , . . . , v n , s& inter variabiles illas proponuntur duae 
aequationes : 
F ( f , , 4 , 4 ) = o , < J >( 4 , 4 , 4 ) = 0 , 
erit 
d$ l 8^ 8^_ 2 ; = + ^ . 0 &A 9^2 • • • 9^n-2 
0F 0$ 0F 0<3> 0v x dv 2 dv n ) 0F 0$ 0F 0$ . 
bL-Mn - 9£»- 1 . 3v»-l0Vn ~ 
His third theorem is a special case of his first, and may therefore 
be passed over. Then we have 
Theorema 4. Supponamus , £ 2 , . . . , £«-i datas esse sub 
forma fradionum 
V + d£»- i - 'V 2% 9^2 
^ ~ 0v x 0v 2 dv n -i u n ^ “ U 9^1 3v 2 
S^n-l 
&„_i 
ubi in altera summa inter indices permutandos etiam referri 
debet index 0 seu index deficiens. 
From this is deduced 
Theorema 5. Si loco functionum u , u 
i ’ 
u 9 
ponitur - , — , ^ , • . . , ^-1 , designante t aliam functionem 
t t t t 
quamlibet, expressio 
du x du 2 du r 
U 0Vj 0V 2 
dv Y 
obit in 
-2 
t n *-* 
0% T 0% 2 du ri 
dv 1 dv 2 
dv„ 
sive in differentiations bus instituendis denominatorem com- 
mumen t ut constantem considerare licet. 
The last of the series is 
Theorema 6. Sint u , u x , u 2 , . . . , u n _ L expressions 
linear es aliarum functionum w , w 2 , w 2 , ... , w n-1 , datae 
per aequationes huius modi 
