8 



SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. I23 



For the reduction and solution of these equations it will be helpful 

 to express them in matrix notation.^^ Thus the set of seven equations 

 represented by (3) above may be written 



L = A 



(4) 



where L is a seven-row by one-column matrix comprising the c values, 



'x' 



and A, the matrix coefficient of 



8 



LCJ 



, is a seven-row by three-column 



matrix comprising the kn, h, and w„ values, in (3) above. 



We should like now to condense with proper weighting the in- 

 formation contained in the several equations represented by (4). 

 The seven ordinary equations contained in (4) can be reduced to 

 three equations by the method of least squares by multiplying both 

 sides of equation (4) by A', the transpose of A. This gives 



A'L=A'A 



Solving for the unknowns 







= {A'Ay^AJL 



(5) 



(6) 



The reciprocal of the matrix A' A is given by the adjoint matrix of 

 A' A divided by the determinant of A' A. Substituting this in (6) 

 yields 



x 

 S 



2id]{A'A)A'L 



\A'A\ 



(7) 



The matrix L alone contains the observed data for the day. The re- 

 mainder of the right-hand side of (7) reduces to a three-row by 

 seven-column matrix that is calculable once and for all, and is at hand 

 for the indicated multiplication. The values for the three unknowns 

 for any set of observations are obtained by multiplying L by this 



ad] {A' A) A' ^. ^, 



■' ^ ^ Smce the seven- 



three-row by seven-column matrix, 



A' A 



row by one-column matrix L consists simply of the seven values of 

 the optical density reduced by the molecular scattering at the particular 

 wavelength (and, at Place 24, by the water absorption if necessary), 



11 See, for example, Perlis, S., Theory of matrices, 1952. Cambridge, Mass. 



