CEETAIN APPLICATIONS TO THE THEOEY OE DEFINITE INTEGEALS. 769 
integration is etFected we must write for <p, and the several rational fractions for 
which they stand and then perform the operation 0, as we are directed to do by the 
definition of that symbol. 
On actual integration we have from (10.), 
2:J^-4/"dT=e[(p]'^“2(^cos-^+x/— 1 sin— j logjx- (cos— +^/-l sin— j-4^”|+we[(p]'^"C, 
the summation in the second member extending from r=0 to T—n — 1. The last term 
m 
in that member is equivalent to an arbitrary constant. For <p and do not contain the 
variables &:c., which are only found in x- Hence the coetficients of terms in the 
m m 
developments of the function (p-v//” will be determinate constants, and C, on 
account of the arbitrary factor C, will be itself an arbitrary constant. 
22. We are thus led to the following theorem; — 
m 
Theoeem. — The value of the expression 2j^'4/"dx, where <p and 4^ cuny rational 
fwnctions of x, and the simultaneous values ofx in the several integrals under the sign 2 are 
m 
determined hythe algebraic equation '4/" = x in which is a rational function ofx, will be 
expressed by the formula 
2j^4z»(?;r=0[?)]-4z«2(cos^+x/--l sin^)log|x-- (cos ^+^/-l sin^)-4/«|+C, 
the summation in the second member extending from r=0 to r=-n—\. 
23. In the particular case in which m=l, w=2, we have 
sjipx/ •4/(?a7=0[<p]\/ log 
X— ^ 4' 
' x+ ^ 'I 
Let us apply this theorem to the problem of Art. 17. We have 
( 12 .) 
2m^;r®cos 
2L a/*— 2m^a?^cos 
= 0 ~ 2wV cos log 
vx + — V ' cos 5 + m"* 
vx->rm^+ V — cos ^ + 'nP 
First, then, we must develope the function in the second member in ascending powers 
1 
x‘ 
of X, and seek the coelRcient of -. Now 
X 
»y m'* — 2mV cos S-\-x*=m‘ — a^^cos 6-\- &c. 
Substituting, the function becomes 
(^— cos^..^log 
vx + x^ cos S . . 
2m^-j-vx — x^ cos 6 .. 
5 H 
MDCCCLVII. 
