COMPUTER-ASSISTED PRODUCT DEVELOPMENT 791 where P•,i indicates the percentage of Ai in X. The percentages must be ex- pressed as decimals in order that the final composition of the paint totals 100% expressed as unity. The matrix Q, dimensions 4 x 6, expresses the composition of the main components A• to A•, each in a row: which represents: 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 q4,4 q4,5 q4,6 A• = B• + [O(B• 4- O(Ba) + O(B4) + O(Bs) + O(B6)] A2 = B2 4- [O(B•) 4- O(Ba) + .... ] Aa = Ba + [O(B•) + 0 .... ] A4 -- q4.4 (B4) 4- q4,a (Ba) 4- q4,• (B6) 4- [O(B•) 4. 0...] S,_'milarly, R is a 6 x 7 matrix expressing the composition of the six compo- nents B in terms of the seven subeomponents C one ean find the percentage of a given subcomponent Cj in a given component B• by loeating in matrix R the element r•,•. The same holds true for the final matrix S, the rows of which express the composition of the seven components C in terms of the 12 ingredients D. If further breakdown steps had been necessary, more matrices could be written the general formula is to put the compound items on the left-hand column and their constituents on the top row. Thus, the matrix will always have the dimensions i x ] where i is the number of com- pounds and i is the number of eonstitnents. It is now necessary to reduce the matrices to a i x 12 matrix that expresses the laequer in terms of 12 ingredients. This is accomplished in three steps. Let T be a I x 6 matrix that is the product of matrices P and Q, i.e., T = P x Q. The lacquer is now expressed in T as the product of six components. Matrix U = T x R is a i x 7 matrix expressing the lacquer in terms of the 7 sub- components, and finally matrix V = U x S has dimensions i x 12 and rep- resents the composition of the paint in terms of the 12 ingredients, D• through D•. If we desire the final composition, we instruct the computer to print matrix V if we desire any intermediate composition we ask the computer to print out the appropriate matrix. V will appear as 12 numbers in .a row, representing the amounts, expressed as decimals, of each of the 12 ingredients. For display pur- poses, we might ask the computer to determine another matrix V • which is the transpose of matrix V. This would cause the data stored in V as a row of 12 elements to be stored in V • as 12 rows of one element. If we ask the computer to print V • x 100, the percentage amount of each in- gredient in the lacquer will appear in a column. The computer then prints the

792 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS elements of a string numbering 12 ingredient names after each row of V • x 100. The output consists of the percentage followed by the ingredient name, one to a line. By making each matrix larger than required, a new ingredient or subcompo- nent can be readily added when it does not occur in the original formula. The chemist needs only to enter zero or the percentage of the ingredients in the appropriate line of the matrix. The formulation of a typical nail lacquer as shown in Table II is rapidly handled by the computer. Table II Computer-Formulated Nail Lacquer (Typical Composition) 0.785481 8.71251 34.4034 15.2004 9.05484 19.8769 9.241 1.7625 0 0 0 0 0 0.705 0.132062 0 2.40428E-2 0 Camphor Plasticizer Butyl acetate Ethyl acetate Isopropyl alcohol Toluene Nitrocellulose Bentone 38 powder Bentone 27 po vder D & C Red •P 7 Ca lake Iron cosmetic oxide Iron blue D & C Yellow # 5 Zr lake Titanium dioxide D & C Yellow •p 6 A1 lake D & C Red •P 9 Ba lake D & C Red •P 6 Ca lake D&CRed #34Calake Do you want ratios and totals or stop? YES Ratio-Modified Resins: Nitrocellulose = 0 Ratio-Gel]ants: Resins -- 0.1907 Ratio-Pigments and Gellants: Resins --_ 0.283871 Ratio-Resins: Solvents: 0.117682 Ratio-Plasticizers: Resins: 1.03145 Ratio-Gellants: Pigments = 2.04679 Total Solids -- 12.3007 Total Pigments -- 0.861105 Total Gellants: 1.7625 Total Solvents: 78.5355 Total Plasticizers = 9.53289 Total Resins -- 9.24224