418 Mr. Adams's Mathematical Problem. [Dec. 



to the sun and the earth, which causes a considerable difference 

 in her brilliancy. 



She is now approaching to the shape of a half-moo)i, and 

 would appear more brilliant to the naked eye, if she were shin- 

 ing with a full illuminated disc. 



Observatory, Gosport, Sept. 22, 1820. 



Article III. 



Mathematical Problem. By Mr. James Adams. 

 (To Dr. Thomson.) 



SIR, Stunehuuse, Sept. 3, 1820. 



If you will be pleased to insert the following problem, Sec. in 

 the Annals of Philosphy, when convenient, you will obhge 



Your humble servant, 



James Adams. 



Problem. — To find the value of , when v iS s,reater than 



w, and m any given number. 



Put V — w = r, then will d" = (m; + r)'" and w" = (v — ;•)", 

 we then have 



V -u) ~ r ~ \ 2 ' to 2.3 ' \» j 



+ &c.) m ?«""-' (E.) 



By substituting for ?<;"*, we have 



tim — to"" t,m _ („ _ ry» / TO — 1 r (m — 1) (m — 2) / r \ 



V - w ~ r ~ \ 2~ ■ V 2.3 ' V v) 



« 



+ &c.) m u"-' (F.) 



If in the preceding equations v — w = r,he supposed indefi- 

 nitely small, then may all the terms affected with -, (-) , (- j ' 



-, (-) , (- j , &c. be omitted as being indefinitely small ; on 



this supposition we should have " ~"''" = m,vf~\ or m ■u""', 



indefinitely near 



The expression given in the problem may be differently repre- 

 sented as follows, viz. 



