## The Algebra of Invariants (Google eBook) |

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aa'u apolar conic apolar form binary forms complete system concomitants condition consider containing the factor contains the term coordinates corresponding covariant of degree cross-ratio cubic curve deduced derived by convolution determinant equianharmonic expressed in terms finite number form of order forms apolar forms of degree four points given Gordan's harmonic harmonically conjugate Hence Hessian homogeneous homogeneous function invariant factors invariants and covariants irreducible covariants irreducible forms irreducible solutions irreducible system Jacobian letters linear combination linear forms linear transformation linearly independent multiplied operator osculating planes pair Peano's theorem pencil plane prove quadratic quadric quantics rational integral function relation replaced represented respect result sextic shew straight line Stroh's substitution suppose symbolical expression symbolical product symmetric group system of forms syzygy tangents terms of covariants ternary forms theorem values zero

### Popular passages

Page 270 - Prove that if three binary quadraties ax2, i^2, cx2 are in involution, then (aV) (bc) (ca) = 0 ; deduce the locus of a point such that the tangents from it to three conics form a pencil in involution, and state the correlative theorem.

Page 353 - Let the letters a, , a, , ... an be arranged in any manner in horizontal rows, so that each row has its first letter in the same vertical column, its second letter in a second vertical column, and so on; and so that no row contains more letters than any row above it.

Page 49 - The difference between any two terms of a transvectant is equal to a sum of terms each of which is a term of a transvectant of lower index of forms obtained from the forms in the original transvectant by convolution.

Page 85 - Theorem will be given in the next chapter. For the present we shall content ourselves with...

Page 49 - Theorem. (1) The difference between any two terms of a transvectant is equal to a sum of terms each of which is a term of a transvectant of lower index of forms obtained from the forms in the original transvectant by convolution.

Page 26 - ... ADMISSIBLE WAVE-FUNCTIONS In the preceding section we have been concerned only with wave-functions of the orbital motion, but the analysis is unaltered if P0 and P contain also spin and charge dependence, so we shall now suppose that P0 is a completely antisymmetric expression; it can be written as the sum of a number of terms each of which is the product of a polynomial of degree N in the co-ordinates with a spin-charge function. The case which arises most often is that in which all single-particle...

Page 170 - If a homogeneous function of any number of variables be formed according to any definite laws, then, although there may be an infinite number of functions F satisfying the conditions laid down, nevertheless a finite number Fl, F2, ... Fr can...

Page 14 - Px — • • • 1 be an aggregate of terms each of which is a product of factors of the type (a/3), then / is an invariant of the quantic.

Page 367 - ... ON THE SYMBOLICAL NOTATION. As we have said in § 82 the notation used in this work is really equivalent to Cayley's hyperdeterminants. The great advance made by the German school lies in the possibility of transforming symbolical expressions, and, of course, in the proof that every invariant form can be represented as a combination of hyperdeterminants. The reader may feel the need of justifying directly the results obtained by manipulating umbral expressions and accordingly we shall indicate...

Page 308 - This method of representing a point by the parameters of the tangents drawn from it to a fixed conic...