OF THE ROOTS OF AN EQUATION AND ITS COEFFICIENTS. 549 
where the dash over S and P indicates that y-a, w—8, .... have been sub- 
stituted fora, 6; ... : 
Hence, after eedhuetiouts similar to those of former cases, 
*S{L(aBy8). (@—e)(@—L)} =6"| (0) , 5(C) , 10(d), 10(e) , (7), - | (30). 
(a) , 5(0) , 10(c) , 10(@) , 5(e) 
, (0), 5(e) , 10(d) , 10(2) , 
, (a), 5(B) , 10(c) , 10(@) , 
Soa mele (b) , 5(c) , 10(d) , (9) 
Les 24 @ 5 5(), 100) , (f) 
Let us consider Srurm’s functions generally. We may write the following 
equations :— 
S.= Q5:— 
= Q.S.—S, ° * * « (31), 
Si=Q;S;—8, &e. 
where each Q is of the first degree in «; and since S§,_, is two dimensions less 
than S,, we can determine the coefficients in the value of Q. Let 
Sy Sn e See see Lk 
pa ea eo Se oe ee : . 32), 
EEN CG ee oe 
on eliminating the coefficients in Q,_,, after substituting in 
Se= Qi-1Sp-1— Sa . : - : (31), 
we have 
re uo aaa Sina ; a 1 = SE, CEO u . (83), 
eT Sgt) 9) Seal 9) Set 
R. yySesths Seta 
and 
Q,31=— mo 2 ama ‘ “ < = : (34). 
FU RGy ee Sey 
ee 
For the purpose of reducing s,, and presenting it as a function of a, b,.... g, 
it will be convenient to put 
7 , uv 
ee Sigh Sean Z= Sait p Oro &e. . : (35), 
ie 
Si gS Sz 
2 
Si 5) Sy 
