1901-2.] Dr Muir on the Theory of Jacobictns. 
157 
Although the general subject here is new, there is a certain link 
of connection with the preceding paper, in that one of the results 
of that paper is employed, and also that Jacobi is using onco 
more the method of ‘generating functions.’ 
Passing over the first two cases, let us note how he proceeds with 
a set of three equations and three variables. As a preliminary he 
introduces after the manner of the preceding cases the determinant 
of the partial differential coefficients, the sentence in regard to it 
being [page 263] — 
“ Ut similia eruamus de tribus functionibus, tres variabiles 
x , y , z involventibus f(x, y, z) , y, z) , if/(x, y, z) adnotetur 
aequatio identica : 
z d- Q'W'fo) - 
dx dij 
+ ztyQW ty) - = 0 
dz ’ 
quam differentiationibus exactis facile probas. E qua, positu 
brevitatis causa 
v = / (*)[>'($ /)■/''( z )- 4 > (? 
+f{ z )W( x )^'{y) - r (y)'p ' { x )\ , 
flnit sequens : 
d/fo'O/V'OO - ■ftW'O/)] , z /WWW - 
dx dy 
, z „ 
dz V ' 
Here the concluding identity has to be noted. He then establishes 
certain results concerning the coefficient of x~ Y y~ Y z _1 in the ex- 
pansion of y , or, as he writes this coefficient, 
[^] x-'y-'z- 1 • 
Thus prepared he attacks the given equations (p. 284) — 
c r = ax +by +cz + dx 2 + exy + . . . , 
t = cl x + b y -f- c z + d x 2 -t- e xy -1- . . . , 
v — ax + b"y + cz + d”x 2 + e"xy + . . . ; 
obtains first the derived set 
