By Warner A.H.
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Additional info for A Comparison of the Thermionic and Photoelectric Work Functions for Clean Tungsten
0 am 0... 0... 0 0 ... ... ... am . ... ... bn A particular case of a resultant is shown in the next paragraph. The degree of R(f, φ) in the coefficients of the two forms is evidently m + n. 4 Discriminant of a binary form. The discriminant D of a binary form f is that function of its coefficients which when equated to zero furnishes a necessary and sufficient condition in order that f = 0 may have a double root. Let m−1 f = f (x1 , x2 ) = a0 xm x2 + · · · + am xm 1 + ma1 x1 2 , ∂f ∂f , fx2 (x1 , x2 ) = ∂x .
Y fg . (90) y=x Special cases of operation by Ω upon a doubly binary form, not a product. In a subsequent chapter Gordan’s series will be developed. This series has to do with operation by Ω upon a doubly binary form which is neither a polar nor a simple product. In this paragraph we consider a few very special cases of such a doubly binary form and in connection therewith some results of very frequent application. )2 . In proof (74), r Ωr = (−1)i i=0 and r ∂r ∂r , r−i i i i ∂x1 ∂x2 ∂y1 ∂y2r−i r r (−1)i (xy) = i=0 r r−i i i r−i x x2 y1 y2 .
2. THE ARONHOLD SYMBOLISM. SYMBOLICAL INVARIANT PROCESSES57 In view of (75) we have here not only the symbolical representation of the transformed form but a very concise proof of the fact, proved elsewhere (Chap. I, (29)), that the transformed coefficients are polars of a0 = f (λ1 , λ2 ) = am λ . The formula (66) for the polar of a product becomes αxm βxn yr = r 1 m+n r s=0 m s n αm−s αys βxn−r+s βyr−s , r−s x (77) where the symbols α, β are not as a rule equivalent. 3 Symbolical transvectants.