Roberts — On the Reduction of an Integral. 99 



or, multiplying both sides of the above equation by v, and dividing by 

 XI + Xv we obtain 



2vdp 2dp mJ dz 



w + Av p^ + A M + Aw \f(%\ 



!N"o-w tT'is of the degree 1m - 2 in (z), and if we take A = - 1 ; w - r, 



is only of the degree m - 1 is z, and consequently consists of a 



u - V 



rational part, the highest power of % in which is 2m- I, and a 



A 



number of fractions of the form , where z - a is a root of 



z- a 



u - V = 0. 



It follows then, by integration, that I„^i depends on /„_2, . . . /o, 



and integrals of the form Z ( , a) when a is a root of u - v = 0. 



"What divisions, then, are we to make of, or into what classes are 

 we to divide, all the elementary integrals upon which the integral 



cf>{z)dz 



depends ? Our division must be distinct. 



"We have already shown that we need only consider the integrals 



lo, /i, . . . I2m-2j and those of the type Zi , w as all others can be 



expressed in terms of these. 



JSTow it is clear Z , w j is an integral which in general it is 



impossible to connect with the I^\ and if we look on X( , n\ as 



belonging to a definite class, we see that we can express the 2m -II, 

 in terms oi particular integrals of the class, and consequently cannot 

 properly regard them as distinct. 



Hence we hold that there is but one class of integrals to which all 



others can be reduced, namely the classZf , w ]• 



There are many other properties of these integrals which might 

 form material for another paper. 



H 2 



,y 



