1909-10.] Dr Muir on the Theory of Orthogonants. 
265 
XIII. — The Theory of Orthogonants in the Historical Order of 
Development up to 1860. By Thomas Muir, LL.D. 
(MS. received October 25, 1909. Read November 22, 1909.) 
Notwithstanding the generalisations made by Jacobi and Cauchy, the 
special case with which the whole theory originated continued from time to 
time to attract attention. In 1843 William Thomson, afterwards known 
as Lord Kelvin, published under the signature “ T.” in the Cambridge Math. 
Journ., iii. pp. 247-248, a short note in which he proved the detached 
theorem that if l x , m x , n x , l 2 , . . . be nine quantities such that 
l\ + m\ + n\ = 1, Z X Z 2 + m^m 2 + n Y n 2 = 0, 
ll + m\ + n\ = 1 , Z 2 Z 3 + m 2 m 3 + n 2 n 3 = 0 , 
l\ + ml + nl = 1 , + m B m 1 + n 3 n Y = 0 , 
then it follows that 
l\ + T\ + l\ = 1 , l l m 1 + l 2 m 2 + l 3 m 3 = 0 , 
m{ + m\ + m\ = 1 , m^a Y + ri 2 n 2 + m 3 n 3 - 0 , 
n\ + n\ + n\ = 1 , w 1 Z 1 + n 2 l 2 + n 3 l 3 — 0 . 
This led to a short paper by A. Gopel in the Archiv d. Math. u. Phys., iv. 
(1843) pp. 244-246. The subject was again taken up in 1848 by L. Schlafli 
in the Mitteilungen d. naturf. Ges. in Bern , Nos. 112, 113, pp. 27-33,* and 
in 1850 by V.-A. Lebesgue in the Nouv. Annates de Math., ix. pp. 46-51. 
Details of these papers need not be given. We may take the opportunity 
to note, however, that after the appearance of Cayley’s paper on matrices 
in 1857 the known general theorem embracing that just mentioned might 
have been briefly formulated by saying that — If MM' = 1, where M is any 
square matrix and M' its conjugate, then also M'M = 1. 
Kummer, E. E. (1843). 
[Berner kungen liber die cubische Gleichung, durch welche die Haupt- 
Axen der Flachen zweiten Grades bestimmt werden. Crelle’s 
Journ., xxvi. pp. 268-272.] 
To prove the reality of all the roots of the equation mentioned in the 
title of his paper — a problem first solved by Lagrange in 1773 — Kummer 
* Published also in Archiv d. Math. u. Phys., xiii. pp. 276-281. 
