98 Mr. Jarrett oh Algebraic Notation. 



and d,""-' .'^^'" 

 5"',.|2m.^^ i-^ .(-!)"■. 2r] .sin 2 rz 



V / tn—r I 



5"' 12 ?« .,^^^ . 271""*"' . sin 2 rz. 



(-4r- '•L^ l'»-^ 



Whence, substituting these values, and putting z = nt, 



QD p2m— 1 ( — iV"' 12m— » . ~~-^— 



u = ntA-S ^-^^ .5'M2m-l/ — .2r-l .sin2r-l.n< 



"^ "'(-4)'"-'. |2>»-l "^L^ p-r 



+ S„, . S'\ .\2vi. f^ '- . 2r\ . sin 2 rnt. 



+ '^"'(_4)"'.l2m L^ [711 -r I 



34. These examples are probably sufficient to show the ad- 

 vantages that may be derived from the proposed notation. I shall 

 therefore conclude with a comparison between it and that at 

 present used. 



1 . P»„ . a,„ = a, . as, . a, . . . a„ ; 



In = n .{n±7-) [n±2r) . .. {n±m- 1 . r) ; 



[n = n («- 1) . . . (n — m+1), and ^ = 1 . 2 .3 . .. «. 



ffl 



3. 5",„a,„ = a, + flj + a., -t- &c. + a„ . 



3. 5'-„A\a„,„ = a,, , + a,, „ + ai,3 + &c. 4- a,,. 



+ «2, 1 + <^-2, 2 + «2, 3 + &C. + 12,. 

 + Oy, I + flj, 2 + 0:3,3 + &C. + 03., 

 + &C. + &C. 



+ flr.i + a,-, 2 + ar,3 + ^c. + a,_ . . 



4. Cr""" . Or = mm of every combination of a, Oj, 03, ... «„, taken 



- 7/1 at a time. 



