PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



51)5 



/=1, 



Also 



^r = ... \/-l cos (^--^ + x^ 

 ('• = 1 ■.. - cos (— + x\ 



cos (—7— + xj = — cos (--J + x\ 



d 



i 



therefore the four values of — i °°® * are 



dx- 



± cos 



(t-)- 



— 1 COS 



(-f-) 



and a trial of any other values of ;■ and / will shew that these are the only re- 

 sults. 



Let 



. . d^ cos X ( > 'TT / =• . TT 1 / 1 \ TT 



Again i — ={cos?- + / . -5 + V-lsm/- + /-^ I cos (»•-;■'+ _l — 



dxi ( 3 3 / V 2/3 



( cos-^ + 's/ — Isin-K-j cos ( — + x\ 



( cos— g- + V— Ism— ^1 cos (-^ + X \ 



(cos^+A/^Tsin^) COS (-^ + ^) 



/ 2-^ / — - . 2 7r\ /TT \ 

 ( cos-g-+V-lsin— ^j cos (— -vx \ 



_cos(|- + ;.) 



/ 2 TT ,__ . 2 TT \ / TT \ 



( cos^- + ^/-lsm-3-j COS (^-- + x^^ 



d «• 



//•=0 gives COS I 



/=0,V=1 ... 



\r=2 ... 



'/•=0 ... 



/=:lJ»-=l ... 



»-=2 ... 



+ a- 



L=2 



27r 

 cos^- 



COS I — - 



/ 4'7r , — - . 47r \ / tt \ 

 ( cos-^5— + V-lsin— ^ j cos (-g- + a: 1 



d ^ cos a; 



which are the nine values of , . 



dx^ 



They may be written more briefly thus : 



COS (— + x\ , -cos(^ + x\, -COS^ — + X \ 



(cos-g-±\/^sm-g j COS (— + x\ , ^cos-g-iA/-! sin^j cos (--^ + x\ 

 and ( cos— ^±\/ — Isin —3^) cos (-^ + x\ 



VOL. XIV. PART II. 



5p 



