P.OO 
PROFESSOR A. R. FORSYTH OX THE 
which is a condition that = 0 and e^{t) = 0 should joossess solutions in common, 
the increased set of five equations is found to be a complete Jacobian system. It 
involves 12 variables, viz., «, h, c, f (j, h, a, b, c, f, g, h, and therefore it possesses 
7 simultaneous independent solutions. 
Two of tlie seven are known ; they are 
A, = + 'Ifjh — af~ — hf — cJr, 
A = he 
where A is the “leadiu^r” coefficient of the contra variant ; two others, bearing to 
the second quadratic the same relation as and A hear to the first, may be expected 
in the forms 
Ao = abc + 21^11 — af~ — bg‘^ ■— ch^, 
A = be - fv 
The whole set of seven, including these four, may be constructed by the known 
processes of solving complete Jacobian systems of homogeneous linear equations. 
Invariants of two Svrfaces. 
19 . Hitlierto we have considered tlie differential invariants which may arise in 
connection with a single function (f) of the independent variables ; and thev may be 
regarded as associated with the single surface (f) = constant (or with the single family 
of surfaces if the constant lie parametric). But in ordinary space, differential 
invariants may arise in connection with tortuous curves or (what is the same thin^ 
analytically) with two families of surfaces such as (f) = consta.nt, cf)' = constant. They 
may even arise in connection with three families of surfaces such as (f) = constant, 
(fi' -- constant, = constant, wdiere there is no identical functional relation between 
(f), cf)', cf)" involving constants only. 
Ihe simplest of such cases occurs AA^lien those differential invariants are required 
which involve no derivatives of a, b, c, f] g, h and only first derivatives of two 
fuirctions cf) and cj). llie number of algebraically independent relative invariants is 4 ; 
and tlie coefficients of tlie highest power of cfi^^^Q which they contain are the indepen¬ 
dent solutions of the comjilete Jacobian system 
c 
c 
c 
0 
y + '' ?1 + -''ig + af 
100 
c 
Ah + / a: + ^ 001 u 
00 
ch 
lOo 
c 
2 / pT + 9 + ^'ooi / 
cb ch . ct • 
V ^ ^ 
cc eg • 0/- ■ - 0<)>Y,^ 
0 , 0 ,, 
o 
001 
?6'\ 
0 , 
0, 
0 , 
0, 
0. 
001 
