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By Roland Deaux
Aimed toward readers unusual with complex numbers, this article explains how one can remedy the categories of difficulties that often come up within the technologies, in particular electric stories. to guarantee a simple and entire knowing, subject matters are constructed from the start, with emphasis on structures relating to algebraic operations. 1956 variation.
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Example text
Z3 - + 23) Z4) (z. + Z4) - (Z3 Z, Z4 Since 2(zl z. - + + (Zl + Z324) = (Zl Z3) (z. + Z4) (Zl + 2. 4 - 2m 34 g If we take the origin of axes at G, we have g = 0, m m2 3 13 m=---. ,GM••) and I GM 1• I I GM.. I I GMI= I GM 12 1 • For another construction of M, see article 104, where M is denoted by O. 22. )z + b,c1 - b1c. = O. 23. If three pairs of points are such that each pair harmonically separates the other two, then the midpoints of any two pairs harmonically separate the third pair. 24. Show that, for any point M, we have (ABCD) = (ABCM) (ABMD).
24. Show that, for any point M, we have (ABCD) = (ABCM) (ABMD). From this show that, in order to change only the sign of an anharmonic ratio (ABCD), it suffices to replace one point in one of the pairs (A,B), (C,D) by its harmonic conjugate with respect to the other pair. Thus, if (AA'CD) = - I, we have (A'BCD) = - (ABCD). 25. Being given three points A, B, C, we construct the harmonic conjugate of each of them with respect to the other two, so that (AA'BC) = - I, (BB'CA) = - I, (CC'AB) = - 1. By using the properties of article 26 and of exercise 24, show that: 1° (ABCA') = 1/2, (ABCB') = 2; 2° the elimination of C gives (ABA'B') = 4 and (AA'BB') = (BB'CC') = (CC'AA') = - 3° from 1° we obtain (ABC'A') = - 1/2, (ABC'B') = - 2, and, by eliminating A, (BB'C'A') = (CC'A'B') = (AA'B'C') = - 1, whence A is the harmonic conjugate of A' with respect to B', C', etc.
Thus we have w-1w = 1, and ww-l = 1. 22. Permutable transformations. Two transformations WI> are said to be permutable if the correspondent of each point Z in the transformation W2Wl coincides with the correspondent of Z in the transformation WlW2' We write W2 W 2W 1 = W 1W2' Examples. 1° Any two translations whatever are permutable. The equation of the product W l W 2 of the translations considered in the first example of article 21 being z' = Z + a 2 + aI' proves, if we compare with equation (16), that Z' is what we wished to establish.