304 
PROFESSOR A. R. FORSYTH OX THE DIFFERENTIAL IXYARLAX'TS 
ihe Jac(>l)ian of one of tlie quadratic forms Avith the other two quadratic forms and 
with the cubic form, being 12 in all. To include the next higher order given bA' 
n — 3, Ave need a cubic form with its set of tAvo associated covariants, a quartic form 
Avith its set of three associated covariants, and the Jacobian of each of the forms 
Avith the originally selected quadratic form, being 9 in all. And so on in succession : 
tlie total number of binariants is 
12+ {(9 +11+ 13+ . +(2n + 3)} 
= IV + 4;n 
With these must be associated the \{n — 1) {n — 2) quantities that do not iiiAmlve 
the derivatives of (/>, these being the Gaussian invariants of deformation ; hence the 
total number is 
+ |n + 1. 
But these are relative invariants ; each of them must be divided by the appropriate 
power of V so that, as one of them is and the quotient is unity, thus making the 
function no longer an iiiAmrlant of the surface, the number of absolute invariants is 
+ f/i 
= In (3n + 5). 
28. Lastly, if Ave require the aggregate of differential invariants Avhlch iiiA-olve 
deriA-atives of E, F, G, L, M, N up to order n — I, and derivatiA*es of tAvo functions 
(f), if/ iq) to order n, the number can be obtained in a similar manner. As in § 27, 
Ave replace the derivatives of L, M, N of the specified orders by the fundamental 
magnitudes of orders 3, 4, . . . , + 1. The algebraically complete aggregate of 
relative invariants of the surface uj) to the orders specified is composed of tAvo 
portions. The first includes the | (n — 1) (n — 2) quantities Avhich do not invoLmthe 
derivatives of (j) and if/, these being the Gaussian iiiAmriants of deformation, as before. 
The second is the algebraically complete aggregate of the system of concomitants of 
a set of binary forms, each divided by a 2 )roper jJOAver of V in order to give rise to 
an absolute invariant of the surface. This set of binary forms contains 
1 ciuantic of order 1, 
4 (juantics . . 2, 
3 . . . 
1 quantic . 
n, 
n + 1, 
being 3a in all. With them must be coiqJed (c<) their (Hermite’s) associated 
coAmriants, the number of Avhich is 
1 . 0 + 4 . 1 + 3 {2 + 3 +. + (a - 1)] + 1 . 7 i 
