78 
MATHEMATICS: F. L. HITCHCOCK Proc. N. A. S. 
From these data the following points are to be noted: 
1. The vital index has the lowest mean value in the quarter ending on 
March 31, the winter quarter. In that period the birth incidence is rela- 
tively low and the death incidence relatively high. 
2. Next in value to this, but standing 9.40 =•= 1.75 points above it, is 
the mean vital index for the autumn quarter ending December 31. The 
difference being 5.4 times its probable error, may be regarded as significant. 
3. The spring quarter, ending June 30, shows the next higher mean, 
being 13.43 ± 1.75 points above the winter quarter. 
4. The highest value of the index falls in the summer quarter, when 
births are most frequent and deaths least so. The mean value, however, 
lies only 5.54 =±= 2.05 above that for the spring quarter, a difference which 
cannot be regarded as significant. 
5. In variability of the vital index, the first two and the last quarters 
of the year, all exhibit significantly the same status. The vital index is 
distinctively more variable in the summer quarter, the difference in stand- 
ard deviations when this quarter is compared with that ending June 30, 
amounting to 6.04 ± 1.45. This may be regarded as significant, being 
4.2 times its probable error. 
It thus appears that the extremely close compensatory relation between 
birth rate and death rate, which Pearl and Burger^ have shown to hold 
in annual figures, does not obtain within the single year. Instead there 
is a well-marked statistically significant intra-annual, or seasonal fluctua- 
tion of the birth-death ratio. 
1 Papers from the Department of Biometry and Vital Statistics, School of Hygiene 
and Public Health, Johns Hopkins University, No. 51. 
2 Cf. particularly Pearl, R., "The Vitality of the Peoples of America," Amer. J. Hyg., 
1, 1921 (592-674). 
^ Pearl, R. and Burger, M. H., "The Vital Index of the Population of England and 
Wales, 1838-1920," Proc. Nat. Acad. Sci., 8, 1922, pp. 71-76. 
A SOLUTION OF THE LINEAR MATRIX EQUATION BY DOUBLE 
MULTIPLICATION 
By Frank Laurkn Hitchcock 
Department of Mathematics, Massachusetts Institute of Techn-oi^ogy 
Communicated by E. B. Wilson, February 28, 1922 
1. Methods of Sylvester and of Maclagan Wedderburn. — A matrix A of 
order A^^ is defined as a binary assemblage of N^ elements a^k, where either 
subscript may have any value from 1 to A^. We add two matrices by add- 
ing corresponding elements. We multiply by the rule 
iAB),k = Sa,A^;(5 = 1, 2, . . . , A^). (1) 
If Ai, A2, . , . , Ah and Bi, B2, . . . , B^ and C are all known matrices, 
while is a required matrix, the linear matrix equation may be written 
