138 MR, JAMES COCKLE; RESEARCHES 
§ 20. 
It is easy to evolve all the roots of 
vu’ — 5 Mut -5Pu}-5Qv’-5Ru+E=0..... (19) 
from a single expression, if unsymmetric functions are 
employed — thus 
{imfli) PME) + OPM) +i%/) +E) 
may, by assigning appropriate values to m, be made to 
yield all the roots in succession; but the possibility of 
evolving them from an expression of the form 
M +2” O'+ 2" Oi + Pit + jer Ov 
where 6', 6%, G1, ©, are functions of the coefficients of 
(19) is by no means an obvious consequence. 
§ 21. 
But, when any one of the six values of @ vanishes, all 
the roots of (19) are included in an expression of the form 
M +776! +70" 418+ m@l” 
where m, as in the last section, is an integer, 7 is 0 or 1 
according to circumstances, and 0’, 0”, 0”, are all known 
or ascertainable functions of the coefficients of (19). 
§ 22. 
If, then, by any practicable transformation of (19), @ 
could be made to vanish, all the roots of the transformed, 
and consequently of the original, equation would be 
determined. 
§ 23. 
There are, however, other modes of proceeding. Super- 
adding relations obtained from other sources — the second 
form of Euler, for example — we may examine the condi- 
tions which the evanescence of @ discloses, and endeavour 
to express w as a function of 6. Thus, in the elementary 
case in which (19) takes a binomial form, assuming 
