71 
coefficients p, q, r, and of the disposable coefficients a, b, c, d. 
He next proceeds to inquire how the three conditions 
a=0...(D) 
b=0...(E) 
c=0... (F) 
can be satisfied simultaneously; and he finds that the first 
(D) is satisfied by making 
3pr/+ 4 q 
a — — — -r — - 
o 
This value being substituted in the second and third equa- 
tions, viz. (E) and (F), and b, c, or d being eliminated, there 
results an equation of the sixth degree. But this elevation 
of degree is avoided by a very simple artifice. Writing 
b — ad-\- and c = d-\- 7 , 
the equation (E) takes the form of a quadratic in d, which 
quadratic is made to vanish identically. The evanescence of 
the first coefficient determines ci, by the solution of a linear 
equation, as a rational function of the known quantities 
p, q, r. The evanescence of the second coefficient, a being 
now treated as known, determines £ as a linear function of y. 
And the evanescence of the third coefficient, when the value 
of £ in terms of y is introduced, determines y, by the solution 
of a quadratic equation, as a function of p, q, r. Hence 
a, £, y are now severally known, and are so determined as to 
satisfy the second equation of condition (E). But it still 
remains to satisfy the third condition (F). Substituting for 
«, b, c their respective values 
Wpl,«d+K,d+',, 
the resulting equation will contain only one unknown, 
viz. d, and a glance at the explicit form of c, given by Bring, 
shows that d cannot rise above the third degree ; so that, by 
the resolution of a cubic equation, d may be determined and 
the last condition (F) satisfied. 
