VOL. LXIX.] PHILOSOPHICAL TRANSACTIONS. ^gl). 



Let n be an even number divisible by 3, then will the above-mentioned quan- 



n 2n zn 

 1^ 



tity A = a"p — L''p" ' — 3a' by + 3a'b 'p = 



In all the preceding cases ;/, in and r denote whole affirmative numbers. 



These equations may be deduced in the same manner as is before given in case 

 3 (1st); or can be demonstrated by writing, in the equation free from radicals, 

 for the different powers of x, their values deduced from the given equation x =: 

 a'!^p + h'^p". To render the solution general, it may not be improper to sub- 

 join the subsequent. 



Lemma. I. Let «, .S, y, S, s, C„ &c. be the respective roots of the equation 

 z" — 1 = O; then will a"' -j- jS" -|- v"' -f- ^^ + s"" + &c. = O, unless n = m, or 

 ?i is a divisor of 7n, in which case a" -{- (3" + y" -|- ^^ -|- i"' -{• &c. = Ji. 



2. The sum of all quantities of the following kind a^jS' + a'(3"' -|- x^y' -\- a-'y" 

 -\- (i"'y'' -j- (i'^y"' -\- ("."S' -\- &c. will be = O; unless n be either equal to, or a 

 divisor of m -j- /•, in which case the sum above-mentioned will be = — n; ex- 

 cept n be either equal to m or /-, or a divisor of them, in which case the sum will 

 be n^ — n; but if m = r, then in the former case will the above-mentioned sum 



= , and in the latter = — -— . 



2' 2 



3. The sum of all quantities of this kind oT^'y'S' &c. + a'P^^'l' &c. -{- 

 at^jS^y'^' &c. -|- x''^'y"'-S' &c. -f- &c. will be = O, unless n be either equal to r -f- m 

 -|- i -)- « &c. or a divisor of it. 



Let n be the number of indices m, r, s, t, &c. and n be either equal to m + ' 

 -|- 5 -f / -f- &c. or a divisor of it, but n be neither equal to, nor a divisor of the 

 sum of any two, three, or four, . . . tt — 3, tt — 2, or tt — 1 of the above-men- 

 tioned quantities; then will the sum above-mentioned = If 1.2.3.4 . . . (tt — 2) 

 . (tt — 1) X n; where it will be -}-, if tt be an odd number, otherwise — . In 

 this case, if a indices be m, b indices be r, c indices be s, d indices be t, &c. 



then will the above-mentioned sum = + r .2.3..« x iS!^ x K2.3!!/x7.2.3..rfx &c. X ^- 

 Let n be either equal to, or a divisor of the sum of any number ^ (less than tt) 

 of the above-mentioned quantities m, r, s, I, &c. and consequently either equal 

 to, or a divisor of the sum of the (tt — ^ ) remaining quantities : find the sum 

 of all possible quantities of this kind 1.2.3. . (j — 2) X (^ — 1) X 1.2.3. . (tt — ^ 

 — 2) X (tt — f — 1 ) X «\ which sum call a. 



Let n be either equal to, or a divisor of the sum of any number (o-) of the 

 above-mentioned quantities m, r, s, t, &c. ; and also equal to, or a divisor of the 

 sum of any number (j) of the remaining quantities, and consequently it will be 

 either equal to, or a divisor of, the sum of the (tt — f' — r) remaining quantities; 

 then find the sum of all possible quantities of this sort 1.2.3. . (<r — 2) X (a- — 1) 

 X 1.2.3. . (c' — 2) X (f' — I) X 1.2.3. . (,r — f - cr — 2) X (tt — f' — a- — I) 

 X n^- which sum call b. 



3 R 2 



