[101 
101] NOTES ON LAGRANGE’S THEOREM. 7 
iables v, v, w ... by 
t may be considered 
sessions (Ixcli/ ... and 
i the functions which 
variables; the former 
to u, v..., the latter 
jquently the notation 
in which expression 8 U , 8 V ... are to be replaced by 
8 U F + 8> u e + 8 U $ ... 8/ 4- 8 v e + 8A .... 
The complete expansion is easily arrived at by induction, and the form is somewhat 
singular. In the case of a single variable u we have □ = 8 U F , in the case of two 
variables, □ = 8 U F 8 F + 8 u F 8 v e + 8^8/. Or writing down only the affixes, in the case of 
a single variable we have F; in the case of two variables FF, F6, cf>F; and in the 
case of three variables FFF, (f)FF, F%F, F9F, FF6, FF<j>, F99, F9<fi, F%9, (f>F(j), 
e determinants. This 
xFcf), <t>F9, 5 where it will be observed that 9 never occurs in the 
first place, nor <£ in the second place, nor 9, <£ (in any order) in the first and second 
places, &c., nor 9, cf), x. (i 11 an y orc ler) in the first, second, and third places. And the 
same property holds in the general case for each letter and binary, ternary, &c. 
combination, and for the entire system of letters, and the system of affixes contains 
every possible combination of letters not excluded by the rule just given. Thus in the 
case of two letters, forming the system of affixes FF, F9, <fiF, 9F, F(f>, 9<fi, <f)9, the last 
four are excluded, the first three of them by containing 9 in the first place or </> 
in the second place, the last by containing 4>, 9 in the first and second places: and 
ling functions become 
. ... dxdy ... 
alue which ^— , 
dudv ... 
lies are changed into 
there remains only the terms FF, F9, cf)F forming the system given above. Substituting 
the expanded value of □ in the expression for F (,x, y...), the equation may either be 
permitted to remain in the form which it thus assumes, or we may, in order to 
obtain the finally reduced form, after expanding the powers of h, k... , connect the 
symbols 8 u e , 8 x f ... 8 U F , &c. with the corresponding functions 9, c/>... F, and then omit the 
affixes ; thus, in particular, in the case of a single variable the general term of Fx is 
far as they enter into 
n order that we may 
V to replace h, k, &c. 
(the ordinary form of Lagrange’s theorem). In the case of two letters the general 
term of F {x, y) is 
hpi-q 
[pfW SuP ~ l8 ° q ~ 1 <t> q Zv0 p 8 u F + 9V8 U <}>18* F] 
(see the Mécanique Celeste, [Ed. 1, 1798] t. i. p. 176). In the case of three variables, 
the general term is 
hpLqJr 
... ' * - s p-^S v-'S r ~ 1 fflp<Wv r 8 8 8 Fa- 1 
As h8 v 9. h8 v 9 ... affect 
[p\P Yqqi \ r\ r ~ J 
by 8 U 6 , 8 V °,..., where 
tiation is only to be 
espectively enter into 
t in the same manner, 
the sixteen terms within the { } being found by comparing the product 8 U 8 V 8 W with 
the system FFF, 6FF, &c., given above, and then connecting each symbol of differen 
tiation with the function corresponding to the affix. Thus in the first term the 
8 U , 8 V , 8 W , each affect the F, in the second term the 8 U affects cj> q , and the 8 V and 8 W 
each affect the F, and so on for the remaining terms. The form is of course deducible 
to multiply this last 
from Laplace’s general theorem, and the actual development of it is given in Laplace’s 
Memoir in the Hist, de I’Acad. 1777. I quote from a memoir by Jacobi which I take 
this opportunity of referring to, “De resolutione equationum per series infinitas,” 
Crelle, t. vi. [1830], pp. 257—286, founded on a preceding memoir, “Exercitatio algebraica 
circa discerptionem singularem fractionum quae plures variabiles involvunt,” t. v. [1830], 
pp. 344—364. 
Stone Buildings, April 6, 1850.