1903 - 4 .] Dr Muir on General Determinants. 
ceeds in a very curious and interesting way. Denoting 
...... . .+/3" 3 a 2 + /3- 2 a + /^- 1 + a- 1 + a- 2 i 8 + a- 3 /3 2 + 
65 
or 
2>-j* 
m=+ oo 
1 
(3 — a 
+ 
1 
since it is the sum of the infinite series for ((3 — a) ~ 1 and (a — /3 )~ 1 , 
he proves after a fashion that its product by (3 — a or a - (3 is 0, 
and that therefore its product by 
1 
1 
y + m(f3- a) 
is simply its product by 
to the product 
1 
+ 
y + m(a- f3) 
Turning then from this lemma 
“o - h 
<1 — 
1 ) 
t Y - U Y / 
where u 0 = a 0 x + b 0 y , u x = b x y + a Y x , he substitutes for the 
first factor of it 
I a (P\ ! x I Vo I d- b 0 (u L t^) I Vo j | a 0 b 1 | x b 0 (wj t x ) 
his justification being the fact that 
V«o - *<>) = KM* - I Vo I + V“i -*i) ; 
but, on account of the said lemma, he leaves the term b 0 (w x - 
out of both denominators. For the second factor there is thereupon 
substituted 
KM 
M I a oh f y - | Vi i } + a i { I a oW | x - | V 0 I } 
+ i V'i i 
M | Vi I - i <*<A I v } + «i { i Vo I - i a A I * } 
* Jacobi writes it — — 4- — - — with the caution that the two parts are not 
0-a a-8 
to be taken as cancelling one another. Of course, also, lower down he does 
not write | a 0 b x | but a - a-J ) 0 or later ( a o&j). 
PROC. ROY. SOO. EDIN. — VOL. XXV. 5 
