1901 - 2 .] Dr Muir on the Theory of Orthogonants. 261 
a + G' V b" V" 
V a - G' c" c 
b" c ' ci' — G c 
b m c" c a" - G' . 
The next and concluding paragraph of the paper is of course 
occupied in showing that by making the substitution whose 
coefficients have just been obtained, the given integral can he 
transformed as desired. 
It is worth noting here that although this paper and the previous 
one are contiguous in the original volume of publication, and the 
problem solved in the second is in essence quite similar to that 
solved in the first, there is not a word to indicate that the author 
viewed them in this common light. 
Cauchy (1829.). 
[Sur l’equation a l’aide de laquelle on determine les inegalites 
seculaires desmouvements des planetes. — Exercices de Math., 
iv. ; or, CEuvres , 2 e ser. ix. pp. 172-195.] 
The equation as it arises with Cauchy would be more fitly 
described as the equation whose roots are the maxima and minima 
of a homogeneous function of the second degree with real co- 
efficients, and with variables subject to the condition that the sum 
of the squares equals unity. 
Denoting the function by 
+ A yiJ y 2 + A Z! z 2 + • • • + 2A xy xy + 2A xz xz + • • • • , 
or for shortness’ sake by s , he of course begins with the known 
equations for determining the extreme values in question, viz., the 
•equations 
0s ds ds 
dx _ dy dz 
x ~ y ~ 7 ~ 
An elementary algebraical theorem gives each of these ratios 
0S 
'dx 
0S 0S 
+ y^~ + + • • 
J dy dz 
x 1 + y 2 +z 2 + 
