Ross — Verh-Fimdions. 



47 



and applying the quotients respectively to 



_1 _1 _ _a 

 ~V "? ~V 



Though some of the exponents of /? are negative, the division is 

 carried out in exactly the same way. Thus we obtain the following 

 values of x from these four new forms : — 

 From the first, 



'J 



which reduces to that given in (2), and has five values. 

 From the fourth, 



-=(-0'^K-ir-(jj(-s--. 



which reduces to that given in (1) and has one value. 

 From the second and third we get, after reduction, 



x = (- af - - - - 4 J' ^ (- a)~'~- R, 



^ ^ 4« 32 ' ^ 64 ^ ^ ' 



which gives four roots of the proposed equation. 



Thus we can deal with this equation in three ways. I^umerical 

 trials must for the present be left to the reader — such cases as 



± 10^ ± 1 = 



being readily worked. In some instances the series will be real and 

 convergent, in others with unreal subjects, or divergent (§ 22), 

 (4) Find solutions for 



We shall obtain 



r -i ^.n-i .1.7: 15 .2.4 



/5 



The example is given to show that fractional indices in the divisor 

 may be dealt with in the same manner as integral indices. But we 

 also have 



R.I. A. PROC, VOL. XXV., SEC. A.] 



