342 
PROFESSOR A. R. FORSYTH ON THE DIFFERENTIAL INVARIANTS 
yve substitute, in each of the arguments such as k', where 
dll 
at 
the proper value of oljtained above for the vaiious aiguments ; we also write 
u' = u -{■ dt 
— 1 + (^10 + '>?oi) dt ; 
and then, according to Lie’s theory, we equate the coefficient of dt on the two sides. 
Tlie functions £ and t] are arbitrary; and therefore, in this new equation, the 
coefficients of the various derivatives of ^ and n] on the two sides are equal. We 
thus obtain a number of partial differential equations of the first order satisfied by/! 
'1 lie construction of the form of f depends upon the manij^ulation of the equations. 
14. The whole process will be sufficiently illustrated in its details if we construct 
the algebraically independent aggregate of differential invariants which involve 
derivatives of two'^' functions ^ and up to the third order inclusive. In order to 
take full account of the increments of such derivatives, it is desirable and necessary 
to retain derivatives of E, F, G up to the second order and, in place of the 
derivatives of L, M, N of that order, to retain the fundamental magnitudes of the 
second, the third, and the fourth orders. Thus the invariantive function involves 
some or all of the quantities 
TT TT 17 77 17 17 
F, F^JI, F.o, Fji, F^J.; 
G, Gjy, Gyp Goq, Gjj, GqoJ 
L, M, N ; 
p, Q, Pv, s ; 
y, S, e; 
^ 10 ; ^ 21 ! ^ 1 - 2 ’ ^03 5 
'Aio> V'on ’/'20> 'Ain ’Ao2> 'Aso 'A2n 'Aj2’ 'Ao3- 
Denoting any one of these arguments by k, the invariantive projierty gives 
_ .. . ) = n-y ( .. . , u ,. . ), 
that IS, 
7 ( • • • > dt, . . ) = 1 1 + (^^0 + ’Iio) dt]->^f{ . . . , u, . . ), 
and tlierefore 
: a« 'dt + ’'■"V 
* The form of tlie results indicates the fonn of the results when more than two functions occur. Morc- 
OA’er, if more than two functioirs of the type of </> and \L lie considered, they are connected hy an identical 
relation. 
