VOL. LXIX.] PHILOSOPHICAL TRANSACTIONS. 493 



X p X [aP"-' + pa'-' + «7°-' + y^i"-' + Py"- ' + y.Q" -' + xS"-' + J.^"- ' 

 + P^"-' + Jp"-' + y.?"-' + V~' + &c. (-7i)] + Z» X i^ X p X [a-p"-^ + 



pV'-^ + ^y-^ + yV-^ + py-^ + y'P"-' + a.'r-^ + -j'^^"-^ + p>-^ 4- 



J2p„-z _|. gjc. (-W)] + C</) X [o^'P'-S + pV-3 + «y-3 + yV-3 -j_ py-3-j- 

 y^P'-3 -f a3(3-3 -|. .f^^-'-s + (3'<J''-3 -j- j3^"-3 + &c. (-«)] -I- dsp X [«*P''-4 + 



j3V-4 + ay-4 + yV-4 + py-4 + /p''-4 + &c. (-«)] + &C. = - vp {aU 



-\- bv + ct -\- ds + &c.) If 71 = 1x, then will the coefficient of h'^p be J-n, i. e. 

 the above-mentioned coefficient will be — np {cm -\- hv -{■ ct -^ ds -\- . . . . -\- ±h^). 



The sum of the contents of every 3 of the above-mentioned values or roots, 

 which is the coefficient of the 4th term of the equation required, will be a^^p^ 

 X [apy -f «pJ + «yJ-|- Pyf + &C. (o)] -h a^:yp' X [«Py' + Oyp'^ -|- (iya' + 

 «P<?'^ -I- &c. (o)] -{- &c. -I- a'^v^yp" X [apy"-" + ayP"""' + Pya""* -{-«pr-^ + 

 ccSfi"-' + &c. (JyJ)] + abi:yp''x [«py-^ + «y'P"-^ + P«y-3 -h PyV-3 + 

 ya^P'"-^ -I- yP^a'-J -j- aP"^J"-3 &c. (1.2.72)] -|- &c. And in general, all the terms 

 (unless the quantity ^p^ contained in the term have this formula ^p" =.p^ or 

 ^^'" = p') will be = O; let the general term be denoted by hlk^p'+:^+' x 

 (a'p-'y" -f- «'P'y" -I- a'^py + "^^P't' + <>^'P''^y'' + '''P'?" + a'P'^'^" + &C.); first 

 let A -|- ju. -|- K neither be equal to n nor 2n, then will the term above-mentioned 

 =: O; if it be equal to n or 2«, then will the term be 1.2 X ?2 X hklp or 

 ] .Inhhlp'. If 2 of the 3 indexes x, (x, v be equal to each other, then divide the 

 above-mentioned term by 1.2; if the 3 indexes be equal, i. e. a = f* = i/, divide 

 it by 1.2.3: find all quantities of this kind where A -|- ju. -|- i, either is equal to 

 n or 2n, and add all the terms thence derived, and call the sum of them a. 



The sum of the contents of every 4 of the values or roots above-mentioned, 

 which is the coefficient of the 4th term of the equation required, will be a* ^p* 

 X [^PyJ-l- apyj -I- &C. (o)] 4- a^b'^p' X [^Py^ + aPy'<^ -\- ^py -j- x'^^yS -f 

 &c. (o)] -I- &c.: let hhlq v^/>' +-"+'+«' X a^P'>'.J^ -|- a'P'^y^^' -|- a'p y-J -|- o^'-^yKSi- 

 + a^^'yS'- 4- a^p«y^-^' + x'^^'^ySi -f &c.) denote a general term ; this term will 

 be = O, unless a + i« + " + i, either = n or In or 3w; in which case the term 

 will be either — l.2.3nhhlqp or — \.2.3nhklgp^ or — ] .2.3nhklqp^; unless a -\- 

 Ij, z=: V -\- I = n, when the above-mentioned term will be — (l.2.3n — ?i^) hklqp^; 

 in this case if a = i/, and consequently f* = ^, then it will be — (1 .2.3ra — 1.2k-) 

 hhlqp'^; but if a = fx = i/ = g = J-??, then will the term be — (1.2.3n — Sra") 

 hhlqp'^. In all these cases, if 2 of the indexes a, ft., v, g be equal, then must the 

 term given above be divided by 1.2; if 3, by 1.2.3; if 4, by 1.2.3.4; and lastly 

 if 2 are equal to each other, and the 2 remaining indexes equal to each other, 

 but not to the former 2, then must the term aforesaid be divided by 1.2.1.2. 

 Find the sum of all the possible terms of this kind, which call b. 



In the same manner from the preceding lemma may be found the aggregates 

 of the contents of every 5, t), 7j &c. roots or values multiplied into each other, 



