108 
Proceedings of Royal Society of Edinburgh. [sess. 
Si~ Ss~ ^6 = ^ J 
02 At b§-\- by = 0 j 
S 3 — So ~ ^4 8" ^6 ~ ^ j 
04 “ A — ^3 8" ^5 = ^ > 
/? 5 - - 6 2 + & 4 = 0 , 
06 “ + ^3 = o ; 
that is to say, provided we have 
^6 = Si S 3 » 
06 ~ ^1 ^3 > 
04 = ^1 “ ^5 » 
b 2 = Si~Si* 
~co 
to 
II 
CH 
1 
O-Jj 
• 
Making these substitutions in 
the original determinant we thus 
know that it resolves into 
x b 1 
-Si x 
and 
Si x Si Si 
& 3 — ?q x 
-ft 
A A 
& 5“A 
- S 3 X 
The conditions for the similar resolvability of the latter factor 
are of course 
Si ~ So ~ S3 ~ Si > ^1 — ^5 = ^3 ~ ^7 5 
Si~ Sz~ S3 ~ S3 5 ^1 ” ^3 = ^3 " ^5 • 
These conditions, however, simply mean that the series 
b 1 , b 3 , b 5 , b 7 and the series Si , S31 Sv Si are ea °h t° be i* 1 
equidifferent progression ; so that if the two common differences of 
the progressions be - d and 3 respectively, the result which we 
have reached takes the form 
x 
-s+ 33 
x b 
I -S X ’ 
b 
x 
-d 
33 
x b — d 
-0 + 28 * 23 
- 2c? it* 
-0 + 3 
& - 2c? 
x 
-3d 
3 
£ 
b-3d 
x b - d 
-0 + 3 x 
x b - 2c? 
-0 + 23 aj 
-0 a* I 
a; b - 3d I 
-0 + 33 x I, 
