1903 - 4 .] Dr Muir on the Theory of Continuants. 
149 
but a preferable is, evidently, 
i * 1 
- n p -'2 
— n+ 1 
2 
P- 4 3 . . . . 
- n + 2 p - 6 . . . . 
-(p -»r*. 
... p - 2w + 2 w 
... - 1 p - 2n 
Heine, E. (1858, Sept.). 
[Auszug eines Schreibens iiber die Lameschen Functionen an 
den Herausgeber. Einige Eigenschaften der Ecmeschen 
Functionen. Crelle's Journ., lvi. pp. 79-86, 87-99.] 
In the case of Heine the functions afterwards known as “ con- 
tinuants” made their appearance under totally different circum- 
stances, viz., while he was engaged in transforming a special 
homogeneous function of the second degree by means of an 
orthogonal transformation. It will be remembered that if the 
quadric 
+ 2 a l2 x x x 2 + 2a 12 x l x z + . . . 
+ a 22 £ 2 2 + 2a 2 ^x 2 x 2> + . . . 
+ Vi + • ■ 
be transformed by an orthogonal transformation into 
■^n^i 2 4- A 22 £ 2 2 + A 33 £ 3 2 + . . . 
the coefficients of the latter expression are the roots of the 
equation 
— A. cq 2 
a \2 ft 22 — 
®13 ^23 
How Heine’s peculiar quadric was 
*bV — % kC o C 1 X 0 X 1 
+ (c x 2 + cf)xf - 2 kc 2 c 3 x 1 x 2 
+ (c 3 2 + c 4 2 )a? 2 2 - 
H” (C‘2cr — 1 4* C , 2cr) X f 
a , 
13 
a 23 
(Xoo - A 
= 0. 
