VOL. LXXIX.] PHILOSOPHICAL TBANSACTIONS. 5^9 



sqx (p+r + s + kc.) . wxjp+q+s + Scc.) . ^. ^„_, , . ^p X {qr + qs+rs+ kc.) , 



sg X (pr + ps + rs + kc.) sr x {pq + ps + qs + kc.) ^^. ^ ^_, _ . sp x (qrs + kc.) 



.gx(prs + kc.) srx(pg« + &c.) ^ ^. ^., ^^^ 



B ^ . . . . 



The law and continuation of this series is evident to any one versant in these 

 matters from inspection. And these fractions may be reduced to a common de- 

 nominator by substituting for sp and a the products sp X v and a X p, where 

 P=fl — r.g'-— J.r — 5. &c. ; for s^- and b the products s^- X a and b X Q, 

 where a=:/) — r./> — ^.r — 5. &c. ; for sr and c the products sr X R and c 

 X R, where R=p — q.p — s.q — s. &c. ; for s* and d the products s* X s' 

 and c X s', where s' = p — q .p — r . q — r . ^c. he. 



The fractions, in particular cases, will often be reducible to lower terms. 



15. Let ?/ = ax^ + bx'' + ' + cx^ + ^^ -j- &c., and the correspondent values of 

 X and y be given as before, then will 1/ = 



*'■ x j' — g' X J^ — r' X x' - a' X gcc. y - , x^Xx'—p^X3* — r^X3*'-^x kc. ^ , t, 

 p^xp'-q'xp'-r'xp'-s'xkc. ^ ^P "T g^ x ?'-;»' X ?'- r* x ?'-*'x &c. "^ ^^ "*" ^^• 



This series may, in the same manner as the preceding, be reduced to terms 

 proceeding according to the dimensions of x ; and the serieses given in the ex- 

 amples may (mutatis mutandis) be predicated of it. 



16. A more general method of correspondent values is given in the Medita- 



, ^, , , x—q.x—r.x-s.kc. , x—p.x—r.x—s.kc. 



tiones- as also the subsequent y = — -— X sfi H ^ -— x s^ 



*• " ' T ^ p — q.p—r.p—s.kc. ^ ' q—p.q — r.q—s.kc. " 



+ &c. as in exam. 1, = s/) + (^-;j)(~-- x sp + ^ x s^) + (jc - p) (a? 



— o) ( — X — X s/) + — X -^ X s^ + — X — X sr) + (^ -_ /,) 

 7/ ^p—q p—r ^ q~P q~^ r—p r — q ' ' \ fj 



(^ - o) (a; - r) (— . -^ . — . X sji + — . — . — X so -I- — . -i- . 

 V'*' 2/ V / ^p—q p—r p — s ^ q—p q — r q—s * ' r—p r—q 



JL X sr -1- — . — . — X s^) - &c. 



r-$ ' s—p s—q s—r ' 



The equality of these 1 different quantities will easily appear by finding the 

 co-efRcients of both, which are multiplied into the same given value of y as sp, 

 so, sr, &c. and the same power of a?; for with very little difficulty they will in 

 general be found equal. 



It is evident from this resolution that, giving the ordinates and their respective 

 distances from each other, the value of any other ordinate at a given distance 

 from the preceding, found by this method, will result the same, whatever may 

 be the point assumed from which the absciss is made to begin. 



k 3. — 1. Let a series be k.x + bo?^ + c^^ -J- Jix'^ -f &c. of such a formula, 

 that if in it for x be substituted a -\- h, there results a series a X (a + ^) + b 

 X (a -i- i)^ + c X (a + h)'' + D X (a -h ^')' + &c. = (as + v,d' + ca^ + Da* 

 -f&c.) X (1 + ^6 -i- ri* + 56^ + /i* + &c.)H-(l + ^a + ra» + ^a^-|-;a* + 



VOL. XVI. 4 D 



