146 Mr. J. Cockle on a Theory of Transcendental Roots. 



as the general equation, the roots of the latter are also functions 

 of a single parameter, and that equation admits of the applica- 

 tion of the processes which follow without any preliminary trans- 

 formation. Retaining the above forms for the sake of brevity, 

 let either of them indifferently be represented by 



and either of their systems of roots by 



<l>l a 9 <l>2 a > • • • <\>tfl> 



no assumption whatever being made as to </>, except its funda- 

 mental property of satisfying the condition 



f<t>a=0 



identically. The next step is to treat this equation by the cal- 

 culus of functions, or rather (since the discoveries of M. Hermite 

 and of M. Kronecker have shown that the roots of a quintic are 

 expressible in terms of elliptic integrals) by the differential cal- 

 culus, which, where the roots are obtainable in terms of integrals, 

 ought to enable us to obtain them. In consequence of its iden- 

 tical nature, the successive derived equations 



df<f>a &fil>a 



da m%h da* - U ' ' * ' ' 



are satisfied identically. Replace <f>a by x in each of these rela- 

 tions, and seek the form of </>. For the quadratic we have 



x 2 — 2# + «=0, 



dx dx 

 da da 



and the elimination of x leads to 



1 C da j 



= — , : x=* 1 — + const., 



2Vl-<z J2</l-a 



= 1— a/1 — a = <t>a, 



the ordinary solution. But here the radical flows from the in- 

 tegral expression. Again, for the cubic we have 



x s Sx + 2a=0, 



whence 



dx 

 da 



