70 Proceedings of the Royal Irish Academy. 



We will nevertheless take it for the subject in order to illustrate 

 the remark in (6), although - 5 is really the proper subject. As f is 

 not small, the approximation is slow. We have (see Table), 



o 2 5 4 ^ 10 ^ 25 



-9 -27+^81-2 243 •••=^•^21 ••• 



from the six terms. 



iN'ext in the derived equation 



23 _ 2^2 _ + 25 = 0, 



where 



the coefficient 15 is nearly as large as 



i 



'\ (25)^ 



while 



^Jr5 > ^25. 

 We have therefore (see Table) where y = -Jl5 



1 1 4 1 25 1 50 16 



x=q+-2+- - — + 



2 8 g 2 f 4 / 128<73 

 taking the positive, and then the negative, value of \/15, namely 



± 3-873 . . ., 



we obtain 



z = 4-237 ... or = - 3-901 . . . ; 



whence 



X = MS . . . ; or = - 1-28 . . . , 



nine terms of the series being taken, and the approximation being 

 very slow. 



The equation - 3:^2 - 2;j; + 5 = is therefore, in this form, a 

 very unfavourable example of solution by operative division, and has 

 been dwelt upon for this reason. But all the roots can be quickly 

 obtained by means of the simple transformation usually employed for 

 removing the second term of an equation — putting x = y + 1 we have 



