of Edinburgh, Session 1876-77. 
391 
and writing 
u = u 2 + u.pc + up? + . . . 
we find 
u = 1 + (2# + 3 x 2 )u + ( x 2 + x 2 )ii ; 
that is 
( - 1 + 2x + 3 x 2 )u + ( x 2 + x 2 )u' = - 1 ; 
or, what is the same thing, 
x 2 ( 1 + x) ’ 
1 
whence 
-3 
u — x e 
but the calculation is most easily performed by means of the fore- 
going equation of differences, itself obtained from the differential 
equation written in the foregoing form, 
( - 1 + 2x + 3 x 2 )u + ( x 2 + x s )u' = - 1 . 
6. On Amphielieiral Forms and their Relations. 
If a cord be knotted, any number of times, according to the 
pattern below 
it is obviously perverted by simple inversion. Hence, when the 
is that of 4-fold knottiness. All its forms have knottiness expres- 
sible as 4 n. 
By Professor Tait. 
{A bstract.) 
free ends are joined it is an amphielieiral knot. Its simplest form 
