693 
OE UO waa ==) ee iel eels lele yy di re ome le (E) 
Under the mentioned conditions the equations (C’), (D) and (£) 
are not only necessary, but also sufficient. In fact, from (£) may 
be coneluded, in connection with (122), since 27, = 2), that V i; is 
symmetrical in / and &, when / and & belong to the same region, 
but j and / do not. From (D) we conclude, in analogical way as 
we have explained in the first part of this paper, that V ij is 
symmetrical in 7 and k, when / and & belong to different regions, 
different from j. (C”) tells that V ij is symmetrical in » and 4, when 
k differs from j. Hence these three conditions are sufficient to show 
that Yi; is symmetrical in the region 1i;, and thus that i; is 
V,,-1-normal. 
When we call’) p,,p,...pq the multiplicity of the roots 
of the algebraic characteristic equation (24), the number of 
equations (C’) is the sum of the two-factorial products of the numbers ° 
Py» Pa ---Pg, and the number of the equations (D) is thrice 
the sum of the three-factorial products of these numbers. The 
number of the equations (£) is equal to the sum of the products 
. + Pk 
of the form pz pr (mae = 1). 
13. Simplifications for the case that the given congruence is 
V,,-1-normai. 
When i, is V,—;-normal, (C) passes into (C,) or (C,), being 
valid for the case that i; and iz belong to different regions. (D) can 
also be brought into the form (D,) and is then valid for the case 
that ij, 1, and iy belong to different regions. 
From (97) follows for the case that i, and i; belong to the 
same region and i; to another: 
ieee ire, Vasu Onset a ey (128) 
This equation can also be written in the form: 
4 
VARS CA OE oe med Nd) 
which has a formal analogy to (D,"), but which is valid under 
different conditions. But the increment of the vector i,, when 
') (0) is the equation (C) of Ricci, Dei sistemi, page 312, but deduced from 
th, and not from V — in 
*) Compare Ricci, Dei sistemi, p. 312. 
4) (1) is (Cy) of Paces, Dei sistemi, p. 314. 
