134 MR. JAMES COCKLE; RESEARCHES 
§ 10. 
When a=; and E=3 
(5) is identical with (1), and (9) with (3). Therefore, de- 
veloping (9) and comparing coefficients, we have 
3a 8-5* F 57 
oy oO sp” Sp ime toes ot. ae 
=o Or VSP. oe toe BF ; a= op? OF a 
§ 11. 
In (1) let Q=O and E=-1 and replace v, w, 2, y, 2, 
by 1, i, ?, @, #, respectively. Then, for all integral values 
of m, f(v”)=0 and @ vanishes. But if we make v=7, w=1, 
the other roots remaining undisturbed, we find 
Ad =ai, f@)=bi', {@) =c?, fU)=d, 
and 6=abedi?=-5*. Consequently the equation in 6 
will, on the present supposition, be 
ETE | hae amaile air (10), 
whence, comparing (10) and (3), 
e= Blo 
and, by the last section, 
§= 2.5% — 23%8e= 2.55? — 2. 3°) =— 58-5". 
§ 12. 
The equation in @ is, therefore, 
65 + 2QE5°4 + 2Q'578 + QH*5"F 
— (58Q°— BP Bb pores 0 | ok cine (11) 
§ 13. 
We may obtain (9) from (5) thus; let 
SGjH=14itPo+ Py +Hz, 
then z, y and ¢ are the roots of 
4 2024 Bu +5=0 ROR otis. (12), 
