170 Proceedings of Royal Society of Edinburgh. [sess* 
respectively, with the result 
dx k 
~ r 
+ 
§g 
dx k ■ 
. — r. 
+ * * • 
+ 
df 
i 
\& 
= A s 
dx 
df 
dx 1 
df 1 
dx n 
df 
df ’ 
A. 
dx k 
—fr 
+ 
A 
. Ar, 
V 1 
+ . . . 
+ 
d A 
fak r 
= 
dx 
A 
dx 1 
dx n 
'S/i ” 
Vi 1 
Vn. 
V 
1 
. A 
+ • * * 
| 
Wn 
II 
dx 
df„ 
T 
dx 1 
Vn 1 
T 
dx n 
it is evident from (a) that on adding column- wise we shall find 
the coefficients of all the unknowns equal to zero, except the co- 
efficient of x k which is unity, and therefore that 
dx k dx k 
n = ¥ s + ¥i Sl + 
dx k 
+ ¥n Sn ' 
Jacobi is thus led to formulate the proposition : — 
“Sint variabilium x, x 1 , . . . , x n f unctiones /, f 1 , . . . , f n . 
a se invicem independentes, si proponetur hoc aequationum 
linearium systema, 
0/ df 
ic r + Lp + 
3/, df, 
iP + + 
v 
dx n 
A, = 
Bx„ ” 
+ = s ’ 
S 1 ’ 
A r + A r + 
dx dx 1 1 
^ V 
+ dxA 
earum resol utio semper est possibilis et determinata eruntque 
incognitarum valores : 
r = 
dx 
df S + 
dx 
¥i h 
+ • • • 
+ 
dx 
¥ Sn 
dx-, 
¥* + 
dx, 
—7 S, 
¥ 1 
+ . . . 
+ 
dx, 
VP 
dX n 
¥ s + 
dx n 
WP 
+ • • • 
+ 
dx n 
¥J n 
Put into the form of a ‘ rule 5 this amounts to saying that any one 
of the r’s is got by taking its ‘ reversed ’ coefficients and multiplying 
each of them by the corresponding s, and then adding. 
