of Edinburgh, Session 1 870-77. 
363 
— (a + 36, 
1 
-6 
- 6 
1 
4a - 
26 
= (a + 36) 
1 
4 a 
-26 
-26 
1 
3 a 
a 
1 
3 a 
a 
1 
2 a 
a 
1 
-6 
- 6 
1 
0 
- 6 
l 
An - 
26 
1 
3 a 
-26 
+ (a 4- 36) 
1 
a 
-26 
-26 
1 
3 a 
a 
1 
2 a 
a 
1 
a 
a 
1 
2 a 
a. 
— (cl + 36, 
1 -6 -6 
1 3a - 
1 2 a a 
-f (a + 6)a(a + 26) , 
for the last two determinants in the preceding line are each equal 
to a(a + 26) . Thus we have 
S, = 
1 -b -b 
1 3 a -2b 
12 a a 
+ 
1 
ci(ci + 6)(a 4* 26) 
a + 36 
S, 
+ 
1 
a + 36 
as it should be. And this method of showing that the validity of 
the fourth case is dependent on that of the third is applicable in 
other case. 
8. Sevenfold Knottiness. By Prof. Tait. 
(Abstract.) 
From the point of view of the Hypothesis of Vortex Atoms, it 
becomes a question of great importance to find how many distinct 
forms there are of knots with a given amount of knottiness. The 
enormous numbers of lines in the spectra of certain elementary 
substances show that the form of the corresponding Vortex Atoms 
cannot be regarded as very simple. But this is no objection against, 
it is rather an argument in favour of the truth of, the Hypothesis. 
