DYNAMIC HAIRSPRAY ANALYSIS 2 5 9 CONCLUSIONS By considering a simple model of fiber bending, with the assumption of a circular cross section of hair and a shape in the form of an "omega-loop," it has been shown that the ratio of hair stiffness of hair bundles with the same fiber volume is equal to the ratio of their respective diameters raised to the second power. The model was confirmed by experimental results obtained by using hair with various thickness such as Caucasian fine hair, Caucasian normal hair, and Chinese hair. For polymer-treated hair, calculations were performed for several model fiber assemblies, and for generalized fiber distributions on cubic and hexagonal lattices by using the parallel axis theorem. It was demonstrated that stiffness depends primarily on the number of layers in a tress. Theoretical predictions were in good agreement with experimental results obtained in dynamic hairspray analysis. APPENDIX The area moments of inertia of circular cross sections were calculated by employing the parallel axis theorem. It states that the area moment of inertia of a body about any axis equals the area moment of inertia about the parallel axis through the center of gravity plus the product of the area of the body and the square of the distance between the two parallel axes (6). For a single circular cross section (6) • x 4 ?r r I•/- 4 P• - 1 X" The distance between x and x' is r the area is q'rr 2. I2/= -•- + ?rr2 r2 + -•- + q'rr2 - • P2 2q'rr 4 4

260 JOURNAL OF COSMETIC SCIENCE The distance between x and x' is 2r the area is •rr 2. q-rr 4 I31-- 4 --+ q'rr2 (2r)2 + -•-+ qxr2 (2r)2 - 4 P3 - 3q'rr 4 -- - 11.67 The distance between x and x' or x" is r or 3r the area is q'rr 2. I4/= 2(__•+•r4)+ 2(__•+9wr4 ) = 21,n. r4 P4/_ I4•/_21 P4 4q'rr 4 X n' th X n ! cross section, n is even the distance between x and X n is (n - 1)r.