Physics. — “On expansions in series of algebraic forms with different 
sets of variables of different degree”). By Prof. J. A. SCHOUTEN, 
(Communicated by Prof. Carpinaat). 
(Communicated in the meeting of May 3, 1919). 
Notations. 
We start from the system S‘?) with the covariant and contra- 
variant fundamental units e; resp. e),4—=a,,..., dn, and the funda- 
mental multiplications A (outer multiplication) o (general multiplica- 
tion) — (alternating multiplication and — (symmetrical multiplication). 
n(n—1) 
ll) * for A= 
KANE = 
0 45 ha je 
A LES . . . . . 
02, C1, = °°, = covariant quantity of degree? (4-vector) = 
Zn 
sv ORE teeta x a= 
@ 4 -+-€),=€),---€3.= contravariant aS » yo » 
Caan 5 ) En » , = 
Ea, +. Ca, —=E= covariant scalar; e ata = E’= contravariant scalar 
n ue 10-43 
xn ea, ---€a E =e e'a,| 
pen ( eycl. 
Bb] ’ er 
“ea, --.ea E= ea, 
. P ° 
By ?-th (procurrent) transvection of m = m,....Mp=M,....Mp 
bd 
Q 
and r’=r’,...r’, will be understood 
PQ 
mr =(m, Ars) (is Ar?) may. ++ mp rig.) - PG *) 
t 
1) See also: “On expansions in series of co- and contravariant quantities of 
higher degree etc.”, These Proceedings Vol. XXII, p. 251—266, here further cited as 
Ey, of which paper this communication forms the continuation and an application. 
3) Ss is found from R* by omission of all quantities that exist only under the 
orthogonal group. See for these systems: Over de direkte analyses der lineaire 
grootheden bij de rotationeele groep enz. Versl. der Kon. Akad. v. Wet. Dl. XXVI, 
bldz. 567-—580; Ueber die Zahlensysteme der rotationalen Gruppe, Nieuw Arch. 
voor Wisk. Dl«XIII, 1919; Die direkte Analysis der neueren Relativitätstheorie, 
Verh. der Kon. Akad. v. Wet. Dl. XII N°. 6 (1919) blz. 29. 
5) The sign . instead of (,)i for the transvections of the theory of invariants was 
first introduced by E. Waetscu. 
