r 4.1 ] 



For if we fuppofe but five different fadlors in both produiis, 

 two of them nrmfl be in one produd, and three in the other, 

 fince neither produdt can confift entirely of equal fadors, as 

 appears from the above, and neither of thofe two fadors mud be 

 taken three times. Let the fadors of one produd then be 



be X- and x + m^', and the fadors of the other x + a, ^ + i. 



x + c,'. The rcfpcdive produds will be x'^ -f 2 mx> + m'^ at*, and 

 + a \ + 2al>c 



_I_ 0/2C 



X* + b x^ , x^ -\- ac^ x+al>c'. That thefe produds 



+ 2C X ^f + ^^' 



-j- c 



may differ only in the loweft term thefe three limitations are 

 necelTary. We mufl make firfl, a + b + nc = 2m ; fecondly, 

 ab + zac + zbc + c^ =m- ; and thirdly, labc + ac'^ + be- = 0. 



From the two firfl equations we have, a- J^%ab -\- b'' -\- \ac ■\- 

 i^bc + 4c' (= 4^') = \ab + %ac + Uc +.4C' v a — 2ab ■\- b^ ■= 



4a + 4^ X c. From the third equation, we have lab + ac + be = 0, 

 2ab 



•.• c =■ 



a + b 



. This value of c being fubftituted in the 



equation a' — zab + b- = 4a + 4i^x c, there arifesa'— 2^^ + (5*- = 



3 F 2 — %ab. 



