This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1888 Excerpt: ...for Vi the letter z, we have exactly the left side of the octahedral equation; the symbols 3-2, &3, as also &i, which I employ in the text, are the well-known Jacobian ones. K We find, further, for the ikosahedral irrationality: (20) z = q-=gT. V L, --CO an expression, therefore, which coincides with 1+22 when the term ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1888 Excerpt: ...for Vi the letter z, we have exactly the left side of the octahedral equation; the symbols 3-2, &3, as also &i, which I employ in the text, are the well-known Jacobian ones. K We find, further, for the ikosahedral irrationality: (20) z = q-=gT. V L, --CO an expression, therefore, which coincides with 1+22 when the term involving q10 is disregarded. The solution of the tetrahedral equation takes a rather more complicated form. We will in this case first replace the z hitherto used by a linear function of z, which vanishes at the summits of W=0, and becomes infinite at the opposite summits of $ = 0. In this sense we write: s = q, tTg + ( + 1) V;(V3 + l) + 2 For the thus defined we have then, first, the equation: (21a) Z =?2 =64, -' We have thus for our three.equations severally determined one root; we obtain the remaining corresponding roots if we substitute in q = e K for-=. the infinite number of values: iK' iK' V-K+d where a, /3, 7, S, are real integers of determinant 1. Here all such systems a, /3, y, S, as coincide for modulus vs, or can be brought into coincidence by means of a uniform change of sign, always give rise to the same root. 8. Formula For The Direct Solution Of The Simplest Resolvent Of The Sixth Degree For The Ikosahedron. In accordance with the particular significance which we attach to the ikosahedral equation, the second of the formulaa (19)--(21) of course has most interest for us. We have already explained that the simplest resolvent of the sixth degree which the ikosahedral equation possesses has been placed by Herr Kronecker in direct relation with the modular equation of the sixth order for a transformation of the fifth order of the elliptic functions (see 15 of the preceding chapter). The for...
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