486 Mr. Moon on the Symbols sin oo and cos cc , 



1 i—iy 

 1-1 + 1-1+ &c. in inf. =—- ^ ^ 



2 2 



= 1 or indifferently. 



It is thus plain that Mr. De Morgan's argument derives no 

 support from the example of the series 1 — 1+1 — 1 + &c. in 

 inf. 



Mr. De Morgan goes on to say, " And in many different 

 ways (some of which are shown in p. 571) sin co and cos oo 

 appear in formulae which can only be made true by supposing 

 them both to vanish. It must also be observed, that every 

 instance in which the case can be clearly tried by an a ■priori 

 method, confirms the conclusion that indeterminateness of 

 value is to be removed by taking the mean of all the results. 



" Two remarkable classes of instances are as follows : — 



" 1 . Take for example « + i^' + cx^ + ax^ + i^^ + c.r^ + &c., 

 or {a + hx-\-cx^) : (1 — o;^). This, if « + i + c = 0, becomes 



: when x ■=.\\ and its value is (Z» + 2 c), or « + i + c 



— r- (i + 2c), or — (3« + 26 + c), the mean of «, a+i, a + 5 



a o 



+ c." In this example Mr. De Morgan assumes the whole 

 point by supposing the series « + 6^' + c^'^ + ax^ + &c. to be 

 =-{a-\-hx-\-cx'^) : (1 —xF). It will be found by actual division, 

 that 



a^hx-^cx'^^ &c. +a.r3« = « + ^^-y + g^' 



1 —x^ 



3^3w+l ^ g^,3n+2 _f. ^^3ra+3 

 j__-3 , 



a-^hx^cx'-V &c. +^^^'^+1 = ^ + ^-^+/^' 



C^.3»+2 ^ fl^3n+3 ^ 6a;3«+4 

 1 -x^ ' 



a + 6^ + ca:2+&c.+c^3«+2^^+^j^^^' 



1 — Xr 



rt^3n+3 ^ ^,^.3n+4 _^ cx^'^H"^. 

 — _-3 ; 



whence, representing the three last series by Si 83 S3 re- 

 spectively, we have, when j;= 1 and a + i+c = 0. 



