/ 



/ 



/ 



18 Schuberfs investigation of Keplei^s Problem. 



If, for instance, we seek the coefficient of e" sin //, RTising 



from e, we shall have a 



ii,m 



l,s 



5^ wherefore 



_^(0) 



It. 10.9.8.7 



1. 2. 3.4.5 



1 



1 



2iM£..,.ll 



)bi'.o-.J2 



, as above (§ IG) 



§ 20. The general term of siu it (E) (§ 12) is = sin i^ ± — • 



Nrp" 



E« 



-^— — [(M-2r+i)"~^sin (72-2rH-i) ^+(?i-22'-/)"~^ sin (?i-2r-i) <«] 



in which we must observe, that 2r, can never become greater than 



? 



M, therefore the first argument, n—trM, is always affirmative, 

 whereas the other, n-2r-ij will often be negative, in which case 

 the product (rz-3r-i}"~* sin {n-2r-i) fjt. will be affirmative or nega- 

 tive, according as the number n is even or odd, because the sine of 

 a negative angle is always negative. Let the coefficient 



t(t-ffc-f {i+k-+2) (i-f 21-— 1) 



be called J^, in which wc must ob- 



^w — » • ^^ • www V « V 9 ■■' 



serve, that J^ is equal to i, and Jo equal to 1 (IS) (§ 13) : then, 

 the srenerai term of x' will be 



+ 



e 



2 



i + 2Jt 



L«. 



Therefore, 



2 



9. 



rx^sint£=4L^'\E« 



2 



t 



J 



A- 



i+2k 



sm 2 ^ 



± 





1 





g \5+2fc+n 



2 



[ ( n-8r +i)"-^sin( w-Sr-H*>+ ( n-2r-i )*->9iD ( «-2r-i )^] ; 



whence the coefficient of e" sin w^ is to be deduced 

 ident, that the first term, sin t>, is applicable only 



But 



lar 



wh 



particu 



m 



a 



whence it follows, that 



Sfc 



d consenucntlv & 



the first term, which 



shall call A^'"\ becomes 





and the coefficient 



J 



7n(-m-|-.s-fi)(T„4.s-f.2) ( „, -}. 2.*— 1 ) 



1. 



2. 



3. 



s 



M„ therefore 



the first 



term is 



-i<V 



\ 



^ 



^ 



