64 



Proceedings of the Royal Irish Academy. 



merely at illustrating some of the elementary uses of verb-functions ; 

 but a few words may perhaps be added with advantage on its appli- 

 cation to the solution of numerical equations. We may infer that 

 such application is often possible ; but must not expect that the 

 roots of a numerical equation can always be obtained with greater 

 rapidity by means of operative division than by the methods of 

 approximation now in use. In some cases operative division will give 

 a very rapid approximation, and in others a slow one ; while in others 

 again the series may be divergent, or the subject of the invert maybe 

 unreal. 



(1) The rational integral equation maybe conveniently prepared 

 for treatment by means of two simple preliminary transformations. 

 For example, let 



ax^ + hx^ cx' dx + e = 

 be the given equation. Put 



y -I ^ 



X = -■ and X = 

 a z 



Then 



j + hy"^ + cay' + da^y + ea^ = 0, 

 f + dz^ + cez' + he^z + ae^ = 0. 



Both of these forms are free from fractions and can be attacked 

 by the same process, namely, by descendiny division. The equation 

 in z will yield the same result by descending division as the original 

 equation would have yielded by ascending division ; that is, after the 

 substitution is made good. (§ 20.) 



(2) The rational integral equation 



+ p_xx'^~'^ +i?.2^'""^ . . . p.m^\X + = 

 can be put in m - 1 other forms by successive algebraic division by 



and each form can be then dealt with by descending division. The 

 subject of the invert of the original form will be J - j9_„„ having 

 7n values ; that of the second form (after division by x) will be 



'""U ~ P-m+ii having )?i - 1 values ; that of the third ""U - p_m+2 having 

 m - 2 values ; and so on. Each of these forms can also be dealt with 

 by ascending division ; or, by putting 



