68 



Proceedings of the Royal Irish Academy. 



and the approximation will be rapid, because the ratio of the weight 

 functions, namely, is comparatively small. Then, either by 



division, or directly from the formula of § 15 (5), 

 2 8 22 13.12 2^ 



2 



Next, putting ^ = -, we have the derived equation %^ + 80e +16 = 0, 



where |;_4 = 80, and p-^ = 16. The first is evidently the predominant 

 coefficient, because 



■80V /16^ 



but, on taking the form + 162"^ = - 80, the subject ^J- 80 is 

 seen to be unreal. 



(6) The equation given in (3) may be called the critical equation 

 for trinomial equations, because it enables us to detect without difficulty 

 the proper subject for the invert. Researches on the similar conditions 

 which must hold for quadrinomials and higher forms cannot, unfortu- 

 nately, be completed in time for this paper ; but so far as can be seen, 

 the trinomial critical equation will roughly serve for the others. It will 

 therefore be used for the following examples ; but, in some polynomials, 

 the first terms of the invert appear to give correct approximations, 

 even though the rest of the series would appear by the test adopted 

 to be divergent. 



The critical equation may be applied as follows to the general 

 equation : — 



x"^ + p.^x"'-^ + p_oX'''-'' . . . = 0. 

 "We first see (mere inspection often suffices) if p-i is greater than 



!i4i?-2, l/-4-i?-3, etc. 



If it is greater than they all, we divide the equation by so as to 

 make -p-i the subject — the invert will give one root. If p^i does not 

 predominate, we try whether p.2 is gi^eater than 



^^i>%, etc. 



If it is, we make it the subject by dividing the original equation by 

 x""'-, the invert now giving two roots. If not, we try with p^ ; and 

 so on. We then apply the same procedure to the derived equation in %. 



