68 
Mr. Ivory on the Attractions of an 
ffy' . dv ! . dm ' . Q (,) = 2* C (,) ,fy' . dy . C'<° : 
to execute the remaining integration we have only to apply 
the method of No. 4 : let the integral y' • df . dm' . Q (i) be 
denoted by 27? x ; then by the method alluded to, 
C (o 
z.\ 6 2 i 
If y be a rational and integral function of p/, y / 1 — p/ 2 . 
cos. m and s/i — p/ 2 . sin. m ; it must be transformed into a 
series of the sines and cosines of m' and its multiples ; then 
/=M (0) + (I-V*) 1 . M (0 .cos.®'+ (1— yy. M (2) .cos. 2=/&c. 
+ {1— [j,") 1 . N*'*. sin. (1 — jj,'-) . N* 2 *. sin. ssy'&c. 
the general term of the series being (t — . M 1 -”' . cos. u~‘ 
+ (1— yy . N (n) . sin. tv®', where and N^ denote ra- 
tional and integral functions of [V ; and here the integral in 
question will consist of as many parts as there are indepen- 
dent functions contained in y\ In order to find the part of 
the integral resulting from the general term, we must mul- 
tiply that term into the expansion of investigated in No. 5; 
and in combining these two expressions we may omit all the 
terms which, after multiplication, would contain the sines and 
cosines of the multiples of m f ; because these, when they are 
integrated with regard to dm', will be of the same value at 
both the limits, on which account- they will produce nothing 
in the value of the integral : this being observed, the only 
term of which it is necessary to retain is that one contain- 
ing cos. n (V— ot), which may be thus written. 
