Chief Justice Cockle on the Conversion of Integrals. 539 

 I shall apply this process to the equation or integral 



(' xv? dv ,„ .\ 



^J.I^Z?' ( 12 > 



wherein V is any function whatever of v. This integral is in 

 one sense a particular, and in another sense a general case of 

 certain definite integrals used by Boole at page 752 of the Phi- 

 losophical Transactions for 1864. 

 8. From (12) we obtain 



♦-♦M-ra* ■:•■ ( 13 > 



d(j> _ v? 



(14) 



(15) 



dx (l-axV) C2 ' 



d(f> _ g(P + i>Q) + aa? a (t;P a --VP-pVQ) 

 dv ~~ {\-axVf ' 



and <j> may be put under the form 



A xvV-ax^vY? 



*= (1-axY)* ' (16) 



9. Let 



/•fa^IHBT ^ 



then we have, availing ourselves of (16), 



d(j> v d(f> acf> df^_ (a + 2)v¥ , . 



dx'~xdv~~ b 'V + dv~'~ {1-axV)*'' ' ' " [ ] 



and further, 



d<h v d</) acj> d ( v*¥ \ . 



which equation, by putting 



is seen to be of the required form (3). 



10. It follows that the differential resolvents of the trinomial 

 algebraic equations used by Boole may be depressed by one 

 order, or by two orders, if we take the form under which Boole 

 exhibited the resolvent. In other words, we may depress by 

 one order the (n — l)-ordinal differential resolvent of a Boolian 

 algebraic trinomial equation of the nth. degree. 



Brisbane, Queensland, Australia, 

 April 20, lSf>7. 



