3 
tive integers and either equal or unequal, and wherein, if 
the differential equation in which x is the independent 
variable is to be linear, we may give b the value unity in 
one term and zero in all the rest. If we so deal with b then 
(2) may be replaced by 
dti ^ . . d n d> df 
i + aF («)«^2 = ^-( 8 ) 
any number of values being assigned to a. Taking the in- 
tegration with respect to u as definite we shall thus obtain, 
in the practicable cases, 
f<pdu 
as the difference of two functions each of the form 
^(x,v)ff(x,v) d x - 
consequently, if y (x,v) be free from x, 
jdv fdu'(f) 
will be given as the difference of two functions each of the 
form 
fdx fdv'r 
and if \p (x,v) be a perfect differential coefficient with respect 
to x then 
fdv fdu'ty 
will be obtained as the difference of two functions each of 
the form 
fdv'r or Jrdv 
In either case, that is to say, if 
fdv fdu‘(\) — fdx fdv(r 2 — rf 
or if 
fdv fdu'(f> — fiji — rfdv 
we may repeat the process of conversion which, tentative to 
a certain extent though it be, may enable us to express by 
indefinite integrals the roots of trinomial algebraic equations. 
My paper — “On Certain Rational Fractions,” in the Mes- 
senger of Mathematics, No. 15, 1868, contains results which 
may be useful in the application of this theory of conversion. 
In the last two equations above given the repetition of the 
process of course takes place on the functions which con- 
stitute the dexters of those equations. 
