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Theor. 5. A linear alternate vanishes when one variable is zero, or two variables are 
equal. It is unaltered by adding to any variable any sum of scalar multiples of the 
remaining variables. It vanishes when two or more of its variables are linearly dependent 
—in particular, when the order of the alternate is greater than the order of the algebra. 
19. It is easily seen that in an algebra of n’t order the general linear al- 
ternate of m’*" order is a sum of algebraic multiples of |” |m_ |n-m independent 
sealar alternates of m’‘" order. 
20. If 6 (p,, po,... pm) bea linear function of m’™ order, then by th 3 and 
note, * Ao and A-° @ are linear alternates of that order. Also we have more 
constants than we need (n”) in order to make either of these the most general 
linear alternate of m’™ order; in fact we have more than enough constants to 
make also * C'¢ or C: 6 the most general linear symmetric of m’™ order. 
21. Inthe use of A(s) it is not only well to note that it is a linear symbol, 
but also that it is commutative with any constant linear symbol, w, of one variable 
(such as S, V, K, in quaternions). In applying A, however, to a function ¢ we 
can not reduce the value of © by reason of any special values of the variables i, e, 
if for special values of the variables we have ¢ — 9’, we do not therefore have 
Ad= ¢q’. 
22. In any algebra of n’** order, we may take the units 7,, 7, 
i Whine oye 
the numbers of n independent directions of unit length (not necessary rectangu- 
Jar). Also, any number p — 2, 1, -+ 2,1, -+..-+ 2n mm where xy, %» ..2n, 
are ordinary numbers) may be taken as the number of a line whose components, 
according to the parallelogram law of addition, arex, i,, «2, %2, .-%n in. Takinga 
fixed origin O, any point P has a definite co-ordinate p, the number of the line 
O P. Any number of independent lines have two orders of arrangement such 
that the interchange of two lines changes the order of arrangement. A change of 
order in the argument lines of an alternate therefore changes its sign. 
. 25. Consider an m-space bounded by the tangential paths of m independent 
differentials d, p, d, p,...dm p. This space may be taken so small as to be ap- 
proximately an m-parallelogram whose r‘® pair of opposite faces intersect the 
lines of d, p and contain the remaining lines through the points of these faces. 
By (r — 1) interchanges the ‘‘r’*®” order d; p, d, p, .. dr—1 p, dr +1 p,--dmp 
becomes the ‘‘15t” order d, p, . . dm p. These interchanges may be made so as to 
leave d; p first. At the initial ‘‘7*"” face d, p is inward, and we have r inter- 
changes from the 7’" order in the differentials exclusive of d; p to bring the 7’ 
order to the first order, say d’, p, d’; p, dm p, wher d’, p——d;, p is outward. 
