484 PHILOSOPHICAL TRANSACTIONS. [aNNO 177Q. 



X — .X — 7 .a — ^.x ~f .Sec. x — a.x — y.x — ^.x — t.kc. , , . , 



u-&.,-y..-^.»-. -l^c: ^ ^ + ^- «.^-y.^-^.^_..^c. X s^ +&C.; and 

 all the terms except the first in the resulting equation will vanish, for each of 

 them contains in its numerator a factor x ~ a = oc — a=0; and the equation 



•II L • — /3 .« — y . a — J'.a — J.&C. . . , , t 1.1 1 



Will become y = - — k ^ x s' = s\ In the same manner bv 



writing (3, y, S, i, &c. successively for x, in the given equation, it may be proved, 

 that when x is equal to j3, y, S, i, &c. then will y become respectively s®, s>', s', s% 

 which was to be demonstrated. 



2. Assume y = ax' + bx'+' + cx'+^' 4- dx' + i' . . . x'' +{"-')' ; and when x be- 

 comes a, |3, y, S, £, &c. let y become respectively s% s', s% s^, s', &c. ; then will 



X' .X' — /i' .X' —y' .X' — ^' .X' — i' Sec. ^ . XT . x' — ci' . .1' — y . x' — i' . X' — i' . &c. 



y a' .«.' — &' .a.' _ 7' . «J _ (?' . a' — t< .cVc. "■" /3' . /5' — «' . /3' — >' . ,2' — ,^' . /3' — f' . &c 



X s^ + &c. 



This may be demonstrated in the same manner as the preceding theorem, by 

 writing a, (3, y, S, i, &c. successively for x. 



Problem. Let there be 7i values, a, |3, y, S, .<, &c. of the quantity .r, to which 

 the n values s% s^, s^, s\ s', &c. of the quantity 3/ correspond; suppose these 

 quantities to be found by any function x of the quantity x; let n, f, o-, t, &c. be 

 values of the quantities .r, to which s'', s«, s% s'', &c. values of the quantity y 

 correspond: for x substitute its above-mentioned values tt, f, (r, r, &c. in the 

 function x, and let the quantities resulting be s'", s', s^, ,y, &c. not equal to the 

 preceding s'', s-, s", s'', &c. respectively : to find a quantity which, added to the 

 function x, shall not only give the true values of the quantity y corresponding 

 to the values a, (3, y, S, i, &c. of the quantity .r, but also corresponding to the 

 values TT, f, (T, T, &c. of the above-mentioned quantity x. 



Assume s^ — s" = t'', si — &• = Tf, *^ — s"^ = t'', i" — s^ = t"^, &c. ; then 

 the errors of the function x will be respectively t-, t, t^, t^, &c. ; and the 

 correcting quantity sought may be 



J - a . ,T - /3 . X — y ■ J — J*. J — t • &C. y. * — i - x — <r .X — T .kc. ^ 



T! — a, ,Tf — ^ .■!! — y .n — S .n — s. &c. if — f.w— (t.t — t. Sec. 



. X — » .X — S, .X — y .X — S .X — i .hcc. X — It ,x — <r .X — r . &c. , „ 



+ -, J rr- X r- X T« -^ &c. 



■ J — «.j_/3.j_y.f_d.f-s. &c. { — 3-,{— <r.J — T . &c. 



.^lit. Let X — a. or — (i.x — y.x — S.x — t.&c. .x—tt.x—^.x — o- .a'— t.&c. = N; 



^ a.TT P.TT y.T S .-IT £. &C. .TT f.TT (T-TT T. &C. =: 11 ; 



. — a.f — P.f — y-f — ^ • ^ — «• &c. .f — TT.f — 0-.^ — T. &c. ^ P ; 

 — a.(r — p.cr — y.(r — S . t — £. &C. . <r — tt.o- — ^ . cr — t. &C. = S ; 



T a.T (3.T y.T J'.T £. &C. .T TT. T f.T (T. &C. = T, &C. ; 



then may the correcting quantity sought be 



N (-77^0 + FiT^) + f^rb) + F(73T) + ^^O 



This problem may be demonstrated in the same manner as the preceding 



<T 



