A Proof hy Elementary Methods. 
These equations are seen to involve coefficients of p 
217 
8 7 
(1) of the form 0— 1 ... v^hich we shall write 08, 17, etc., the 
index of q being half the difference of the figures, and 
8 7 
(2) of the form — — , ... which we shall write "8., 'Is, the dot 
\ / ^5' ^6' 5' 
indicating zero here and in vacant places in the following 
determinants. 
From the seven equations above p^, p^, ... Po can be eliminated, 
giving the determinant equation 
0 
1, 
2, 
3, 
4, 
5, 
68 
0, 
1, 
2, 
3, 
48, 
57 
0, 
1, 
28, 
37, 
46 
08, 
17, 
26, 
35 
•74. 
06, 
15, 
24 
86, 
•53, 
04, 
13 
•76, 
•65, 
•54, 
•3., 
02 
0, 1, 
2, 
3, 
57 
0, 1, 
2, 
3, 
48 
., 0, 
1, 
28, 
46 
. , 0, 
1, 
28, 
37 
08, 
17, 
35 
08, 
17, 
26 
-8,. 
•74, 
06, 
24 
-83, 
•74, 
06, 
15 
%, -73, 
•64, 
•53, 
13 
•86, -73, 
•64, 
•53, 
04 
Moreover, the middle five equations (omitting the first and last) are 
linear equations for ^3, p^, 2^1, Po, from which we get 
Po 
Calling the first determinant and the second pair and we see 
that if 0^ = 0 gives us a real value of 5, i^^^^" gives us a unique 
corresponding value of p. 
§ 3. We proceed to prove that = 0 has real roots in q (in the result 
we find that there are always 4 + roots). 
(i) The term independent of q is 07 in the leading diagonal. 
* This eliminant can also be obtained by assuming the quotient of the original division 
to be 'boX^—^-\-hj^x^—^...-\-hn—-^x-^hn—2, and determining the 6's ; or, as the eliminant oif{x 
nd/Q. See Dr. Muir, in Proc. R. S. Ed., xxi., p. 360. 
