354 



REPOKT — 1869. 



It follows from (23) that 



—Jlx +y) _/'' F'Ay + /.Vg'.y/'-'^' 



go gxgD^fxfijhxhy' 



Hence 



go—hof(x-\-y) __ {gy—fx hy){gx—hx fy)^ 

 go + hof{x +y) {gy +fx hy){gx + hxfy) ' 

 But from the identical equation 



go-—ho^fv- + (go^fiv"—ho-)fy- 



=go-—ho\fy- -\-fx- (go-fy-—ho-) 

 we obtain from Section S (y) 



gx--ha-fy-=ffy"-f.r- Inf, 



or 



gx — hx fy _ gy—fx hy 



and hence substitutiug 



go—^'ofi-'^+y) \ _ g^i— f'-^\ fy _ gy — /'• Ay . 



go + liot\x +y ) I yy +fx hy ~ rjx + hxjy ' 

 and similarly 



g'^-gj^+y ) 1 _ fahy^-hxfy ^ gil-g x 

 go -^g{x + 3/) J gy ^gx fx hy - hxfy ' 



ho — h{x-\-y)\ _f^ gy-^-fygx hy—hx 

 ho-\-h{x^y)\ hy + hx ~ fxgy—gxfy 



Section 10. — We now come to an entirely distinct scries of theorems, Avliich 

 constitute the most important and interesting addition which has been made 

 to this division of our sxibject since the publication of the * Fundamenta 

 Nova.' 



It is known that the decomposition of algebraical fractions leads to the fol- 

 lowing proposition : — 



^/ 

 v/ 



sin (x—a) sin (x—h) 



sin (a — «)sin(a — 6) 



sin (x—a.) sin {x—j3) sin (x—y) sin (a— /3) sin {a.—'/) sin (x — a) 

 sin(/3— «)sin(/3 — /)) 1 , sin (7—0) sin (7—6) 1 



+ 



+ 



sin(/3— a)sin (/a — 7) sin (.)■— /J) ' sin (7— a) sin (7-/3) sin(.t7— 7)' 

 Similar reasoning, when applied to tlie expression 



leads to the theorem 



F(x) - gi(^--'-'>^i(A'- '^')Q,(p— »•) 

 O^(a~x)0^iro-x)0fy-xy 



d\(o)e<'F{x) = ^'(^- «)^i(^ -a)(^,0>-«) V " _fl' (et^+^ 

 6li(/3— a)ai(7— a) "-'^ sm(a.—x + sri 



) 



_^B f\-l3)d, (^i-i 3)eff,-i3) ^'- q^ei^-^S+mi I 



0,(«-/3)r^,(7-/3) *"-"sin03-.r+,s,;; f 



_^ 6),(\-y)«,(^^y)fl,(p_y) ^- . /e(-^'^+y)' I 



(A) 



