:? 



so Schuherth' investigation of ICepler's jHrohJem. 



§ 22, We have yet to determine the number Jc, If we call 



A^ the sought coefficieut of the proposed term e" sin ?w m^ we shall 

 have the eqciation (§ ^0) 



riZT* ^'^ ( 2 ) • [(n-3r-Fz)"-^ sin {n-2r+i) ^ + (n-3r-£)"-^ sin (n 

 = A^'^e^sinm^; whence it follows, that i + 2]c + n must be 



ri -■ 



9 ' 



equal to a, or^ = ~|^- Substituting, then, the three values of 

 r (§ 31), we shall get. 



^ 23. All those quantities, (&) (§ 19), (c) (§ 20), and [d) (^ 2^), 

 being collected, the total coefficient of e" sin m^ will be found 



1.2...a\.2J "^ 2«-\Ni 



in which I shall again observe, that s = ^ (^ 18), 



'• 2- sTT :— i ■ (i 20), 



vr _ "('■•— I ) ("— 2) ('<—'■+ 1 1 



an,l tUat £ and . signify a„^ integer affi^ative numl.e.-, wherea, 



«>e case t = „ is co„talaed i„ tUe first term A., a,.,l „ = „ ;„ tl.o 



torn of the coefficients J and iV, tj/r, a~i~-n «4.; ^ • 



Blast always be affirmative number f 1^ ^ """"' "~^""' 



numbers : for, when any one of these 



r 

 J 



1 



.5r-2>] 



