128 
THE APPLICATION OF ALGEBRA 
before) the sum of z and -^-, — .9; then, their rectan 
gle being l, the sum of the rcth powers (~ n P —~ 
will be had in terms of .9 ( from Problem 68), •which 
sum let be denoted by S ; so shall our equation becoirte 
: whence the value of s may 
s -1-1 
may, in any case, be determined. 
Thus if (>?) the given number of terms be 3 ; then 
Sr (the sum of the cubes of z and —) being r= s 3 — 3a,. 
we have by.- 
• « • • - - ç- 
is, by division, 6 X a* p 2a 4- I = a? x - 
If the number of terms be 4; then will Sr/ — 
s 4 —4a* P 3 
a 4- 1 
4 a* P 2 ; and therefore b x 
which, by an actual division of the numerators, is re 
duced to b x s 3 4- 2a* — a 3 x s 3 — a 1 — 3a 4- 3. 
Again, taking n — 5, we have S := a 5 — 5s 3 -p 5.9; 
a 5 -5a 3 f 5s— 2 
, a 5 —5s 3 4- 5a -p 1 
a 3 x 
and therefore b x 
which, by division, is reduced to b x s 4 4 2s 3 —a 1 —2a p1 
rr a 3 x s 4 — s 3 — 4a 2 p 4a 4- 1 : and so of others; 
where it may be observed, that the values of S — 2, 
and S + 1, will be always divisible by their respective 
denominators; except the latter, when n is either 3, or 
a multiple of 3. 
PROBLEM LXXV. 
The sinn of any rank of quantities (a b p c 4- d 4- 
e 4- &c.) being given — P, the sum of all their rectan 
gles (ab 4- ac 4- ad &c. 4- be 4- bd &c. 4- cd &c.) 
n Q, the sum of all their solids (abc 4- abd 4- nbe &c. 
4- acd p ace ¿cc. p bed &c. — 11 &c. &c. it is pro 
posed to determine the sum of the squares, cubes, biqua 
drates, &c. of those quantities.