of Edinburgh , Session 1873-74. 
381 
A general theorem on this transformation is given by Stern in 
u Crelle’s Journal,” vol. x. (1833), p. 267 ; and by a quite similar 
method several allied identities have been reproduced in a paper 
recently read before the Mathematical Society of London. All of 
them, however, may be established much more easily by means of 
the above results. Thus — 
1 + 
d l 
(d. 2 ~ e 8 ) d x e x 
dft^ — e 4 e 2 — 
(di ~ e x ) (d, 6 - e 3 ) d, 2 e a 
df , ~ e, 2 e 6 
(d. 2 - e a ) (d 4 — e 4 ) d z e it 
d.d, - e e , — ,, 
/ d l — ey — ( d 2 — &^)dy&y — (d y — ey) ( d 3 — e 3 ) d 2 e 2 
K \1 dyd. 2 e-yG,. 2 d. 2 d z e 2 6 3 
- {d 2 — 62) d\Cy 
— (^1 (^8 ^ 3 ) ^ 2^2 
K (e* dyd^ - c x e 2 cZ 2 e 3 - e 2 e 3 
_ . d 2 {d 1 - e t ) . ^ (d 2 - e 2 ) . . . . . 
. e 2 (c?! - ei) . e 3 (rf 8 - e. 2 ) 
_ dyd 2 d s .... 
e i e 2 e 3 • • • • 
which is the general result obtained by Stern, and which probably 
includes all the others. 
I am indebted to Professor Cayley for the remark that any con- 
tinuant may be expressed by means of a simple continuant. Thus 
dividing the second column by the first constituent of it which is 
not zero, and then multiplying the third row by the same, and so 
on through the remaining columns and rows in succession, we have 
o x by 0 0 0 
- 1 a 2 h. 2 0 0 
0-1 a 3 b 3 0 
0 0-1 a 4 b 4 
0 0 0 - 1 a b 
oj 1 0 0 0 
-1 4 * 0 0 
0 -1 a ir 1 0 
0 0-1 a d)ydi * 1 
x bjb A 
0 0 0 -1 
bybz 
' J zb 2 bi 
