PHILOSOPHICAL TRANSACTION'S. [aNNO 1779. 



In the same manner let n be either equal to, or a divisor of the sum of any 

 number (t) of the above-mentioned quantities m, r, s, t, &c. ; and similarly let 

 n be either equal to, or a divisor of the sum of any number {t') of the remaining 

 quantities; and also let n be either equal to, or a divisor of the sum of any 

 number (f) of the remaining quantities; then will n be either equal to, or a 

 divisor of the sum of the {-rr — t — o-' — f) remaining quantities: find the sum 

 of all quantities of this sort 1.2.3. . (t — 2) X (r — l) X 1.2.3. . (o-' — 2) X 

 {/ — ]) X 1.2.3. . (e" — 2) X (?" - 1) X 1.2.3. . (tt — T — 0-' - e" — 2) X 

 (tt — T — (t' — ^" — 1) X n\ which sum call c, and so on; then will the above- 

 mentioned sum a^p^'J' &C. + a'{i'"y'S' &C. + a'"(i'-y'S' &C. + oc'-(i'y'"S' &C. + &C. 

 = + ( 1 .2.3. .(tt — 2)x(7r— ])Xn — A-fB — c + D — &c.) where it will 

 be + if TT be an odd number, otherwise — . In this case, if a indices be 7n, b 

 indices be r, c indices be s, d indices be t, &c. then will the above-mentioned 



_ 1.2.3.. (2^ — 2)x (t — \) .n — A + B — c+D— &:c . 



sum— 4- 1.2.3. .ax 1.2. 3. .Ax 1.2.3. .ex 1.2.3. .rfx&c* 



7. Let a, |3, y, S, f, &c. be the roots of the equation z" — I = O, and the 

 resolution h& x = aj^p + bj^p"" + c^p^ + d:^p* + h^p"-. . . + A^jb" 



.... + i;yp\.. jf.q"vP^---- + r::/p^.... -k-sj^p-^-Jr ^;^r-3 + t;^/>"-^ 



+ ujyp"-' ; then will the different values of a' be respectively a^p X « + b^p' 

 X a'+ c:^p^ X a^+ d;;/p* X a^ + h^p^X xK.. + k^p'X a^^ 



+ S^P''-iX a"-4+^^/,"-3 X a"-3 + V^p"-^ X a"-^ + M^p"-> X a;"— ; 



a^P X^ + b;^f- X ^' + c^p'x ^^+d^p'X (i\.. + h^p' X P\..+ 



k;yp" X p" + s:yp"-^ x i3«-4 + t^p^-s x p'-^ + v^p"-^ x P"-^ + 



M^/>""' X (3"-'; 



a"^p Xy+bJ^p'^Xy'' +c"^p' X / + d^/p' X y^ . . + h^/p^ X y\ . . + 



li^P'" X y^ +s^p"'^ X y'"^+t:yp"-i X y"-^ +v^lj"-^ X y"-' + 



?i^p"-' X y"-'; 



a^JO X <r + i;^p' X .J' + c^/)^ X S' + djyp' xr .. . + h^p' X s^...+ 



kjyp-' X Si^ + sj^p'-'i X S"-* + t:/p'-i X S"-i + v^p—^ X S"-^ + 



u^p"" XS"-'; 



&c. &c. &c. &c. 



and consequently the sum of the values or roots, which is the coefficient of the 

 2d term of the equation sought, will be a .;>/> X (a + [3 -f y + <J + &c. (o)) -f- 

 b"^p' X {«.' + P'^ + / + ^'' + &c. (o)) + c;./,^ X («^ + P^ + y^ + c^ + &c. 

 (o)) + .... + t^;/j!>'"' X («"-^ + P"-^ + y"-^ + <?"-' + &c. (o)) + w;//^"-' 

 X (^"-' + P"-" + y"-' + S"-' + &c. (o)) =0. 



The sum of the products of every 2 of the values or roots, which is the co- 

 efficient of the 3d term of the equation sought, will be d^ 1/p^ X (aP -}- «y -{- 



Py + «i -1- p^ + y^ + &c. (o)) -f abyp^ X («P' + P*' + n' + y^' + P/ + 



yj32 ^ aJ^ + ^oc -f &c. (o)); and in general all the terms will be O, unless a X m 



