Some Invariants And Covariants of Ternary Collineations
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23, p. 431.
flioa oit, p. 275.
Digitized by Google SOME INYABIANTB AND C50VABIANTS OF TERNARY COLLINEATIONS. 9 be normal. Hence if aa and fib operating on a certain tetrad and counter- tetrad give a 4-point a^ and a 4-line fi., the collineation which transforms &« (the counter-tetrad of 13.) into a. will be normal. According to (9) the collinea- tion which carries 6^ into a. is Therefore the condition that the collineations aa and 8b be apolar is (11) 2:(a,a,aJ(^,^.^,)(o^^0 = 0, where a^ and 13...^ correspond respectivelj through aa and ^6 to a tetrad of points and its counter-tetrad of lines.
It is to be observed that (11) is linear in each of the quantities a. and fi^.
Hence if all of those quantities except a point a^ are given it will lie on a defi- nite line, and if all except a line )8^ are given it will pass through a definite point. Therefore a 4-point and 4-line subject to the condition (11) determine a tetrad of lines through a^ and a tetrad of points on fi^, consisting^ in fact; of the evectants of (11) with respect to a^ and 0..
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To quickly assess the difficulty of the text, read a short excerpt:
23, p. 431.
flioa oit, p. 275.
Digitized by Google SOME INYABIANTB AND C50VABIANTS OF TERNARY COLLINEATIONS. 9 be normal. Hence if aa and fib operating on a certain tetrad and counter- tetrad give a 4-point a^ and a 4-line fi., the collineation which transforms &« (the counter-tetrad of 13.) into a. will be normal. According to (9) the collinea- tion which carries 6^ into a. is Therefore the condition that the collineations aa and 8b be apolar is (11) 2:(a,a,aJ(^,^.^,)(o^^0 = 0, where a^ and 13...^ correspond respectivelj through aa and ^6 to a tetrad of points and its counter-tetrad of lines.
It is to be observed that (11) is linear in each of the quantities a. and fi^.
Hence if all of those quantities except a point a^ are given it will lie on a defi- nite line, and if all except a line )8^ are given it will pass through a definite point. Therefore a 4-point and 4-line subject to the condition (11) determine a tetrad of lines through a^ and a tetrad of points on fi^, consisting^ in fact; of the evectants of (11) with respect to a^ and 0..
What to read after Some Invariants And Covariants of Ternary Collineations?
You can find similar books in the "Read Also" column, or choose other free books by H B Henry Bayard Phillips to read onlineMoreLess
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