540 J. D. HAMILTON DICKSON ON RELATIONS BETWEEN FUNCTIONS 
1,-5,10,-—10,5,-1 
-6b, c,d, ef, g=@3{(a-By ! ; 0 + (7). 
Bla, Mba gs ait i, ytotet+é 1 | 
yo+yetyo+5e+864 6 , yto+tet+l . 
de + ydE+yeb+5e6 — ,yd+yet+yS+de+86+ eb | 
roel , yOetydl+rye6+ bel | 
0 ; de | 
III. To extend this method, I write, for example, the symbol— 
a, Boa lo 1 5) 1 ’ 1 p) LL 
a,B-,¥,8 | Br yt+8\, at ytd 5 e+ P40 [at ery 
1,1 ,1,1 |By+Bd+y6 , ay+ad+yd , aB+ad+ 86 , aB + ay + By 
Bys ; ayd ; ad . aBy 
understanding by it that only three horizontal lines from the second (the square) 
matrix are to be used at once. The resulting determinants are respectively 
equal to 
C53 oy 1 : 1 J 1 i 
a,B,y | Bt y+8 , aty +8 , a+ Bt+d 
1,1 ,1 | By+Bd+y6, aytad+yd ,aB+ ad+ £8 
Bys ; ayd : ad 
a 
== {(a—B)*(a—y)"(B—y)’. 
This also is equal to 
| PP, —2P,; PjPs=3P, 5 Pybs— 42, eae 
| P, ? 22 ) 3P, ) 4P, | 
| 4P, ? 3Py ) oP, ) li 
where those three columns alone are to be considered at one time, which cor- 
respond to the three lines considered at first. This may be written 
1P, , 2P,, 3P,, 4P,, 0 
iP, gee Py, Py 
-, PY Se SP,, 4P, 
) “up APSR, 4 BP, 9 Py 
