By Jean Michel Lemaire

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**Extra info for Aigebres Connexes et Homologie des Espaces de Lacets**

**Example text**

For the case of quaternions, if we suppose i0 = 1, i1 = i; i2 = j; i3 = k; (278) the symbols i, j, k being still connected by the fundamental relations (a); the six symbolical equations (228), and the sixteen symbolical equations (239), will then be included, by (269), in the formula (273), in which we may write, by (240), and by (271), or (238), q0 = a + bi + cj + dk; (279) and the expression (255) will be included in the more general expression (276). And if we farther particularize, and at the same time simplify, by adopting, as we propose henceforth to do, the values (241), which reduce q0 to 1, we shall then obtain from (276), by (278), the same expression (254), or (c), which has already been assigned in the twenty-seventh article, as the representation of a numeral quaternion.

W = ww − xx − yy − zz x = wx + xw + yz − zy (301) It thus appears immediately that w =w ; (302) and the elimination, above directed, of the four numbers w , x , y , z , that is, of the constituents of the numeral quaternion q , between the eight equations (300), (301), gives these three other equations, which complete the solution of the problem, so far as it depends on the above-mentioned elimination: wx + zy − yz = wx + yz − zy ; (303) wy + xz − zx = wy + zx − xz ; wz + yx − xy = wz + xy − yx .

Nn−1 q); (208) which are analogous to those marked (10) and (11), for moments and momental sets, and also to the formulæ (57), (58), for constituent ordinal relations, and for the ordinal sets to which they belong. We may then substitute for the formula (153) of symbolic multiplication, or of successive derivation, the following: ns . ×r ×r = nr,r ,s ; (209) which will give, also, by suitably changing the letters, ns . ×s ×t = nt,s,s ; 32 (210) the commas in the indices being here, for the sake of greater clearness, restored.