acted on by small Forces. 209 
The p+ 1" term of 
(c cos. 0+ a cos. ¢ sin. 0)”. cos. 0\".sin 4 is 
m.m—1. ..m—p m—p+1 
1.2...p c”-P .a? .cos. p\? .sm. gnc cos. 0\°"-?. 
Now when p is odd 
—— : 1 —— 
Josin. 0? +'. cos. 6\?"—? = sin. 6+”. oe COS. O\ae 
m+ 1 
2m—p—1 aM ee eee 
es 2m—p—3 ——$— ———————————_ 
PT 2 Sa ee ey + &e. Tome L.2ma1 Some pt+4.pt+2 
This vanishes when §6=0; when 6= 9, every term involves an 
even power of cos. ¢, which is multiplied by cos. ¢\’, and therefore 
vanishes on being integrated through a circumference at the next 
operation. No term, therefore, results from the odd values of p. 
(12.) When p is even, f, sin. 6\?*'. cos. 6"? is sin. 0? ** x 
1 z 2m—p—1 
ee prea eee eee _O\m—P-3 &e. 
= + 1 COS. 6 SoS Ser Sy i aae cos ar 
2m—-—p—1.2m—p—3....5.3 
2m -+-1.2m—1...... p+to.p+s 
_ 2m—p—1.2m—-p—3......5.3 
2m+1.2m—1.....p + 5.pt+3 
| eee 
. COs. 0 fen sin. 6\? 
+ Pet pai sin. 0\?-*? + &e. 
‘Def Oe OLE! \ 
Pp ciara 3.1 
When 0@= 9, every term involves an odd power of cos. ¢, and 
vanishes on integration with respect to ¢. When 6=0, the expres- 
sion becomes 
_2m—p—1.2m-p—3....3.1.p.p—2...4.2 
2m+1.2m—1..........3.1 
2m—p—1.2mM—p—3....3.1.p.p—2....4.2 
cy) Ee ee ; 
Vol. Il. Part I. Dob 
or the integral is 
