I 396 ] 



Bl't Dr. rialley has already Sufficiently explained this method 

 of applying the fcries. 



To come then in the next place to produds of three dimen- 

 fions, Of which confift each of three fadors — 



Let x'+qx-+rx + s reprefent one of fuch produds, and 

 let its fadors be denoted by a- + a, x+ d and x + c. Then if 

 the fign of the fccond term in each fador be changed, the 

 lign of the firft term in each being unvaried, the produd will 

 be A," 3 — qx- + rx — s ; make q, or ti+b + c=Oj we then fliall 

 have two produds differing from each other by 2s — zabc ; fo 

 that if c be taken equal X.o a -{■ b with their figns changed, 

 the produds will differ from each other in their loweft terms 

 only ; but this difference, and at the fame time the number of 

 different fadors, will be the leaft poffible (fradional numbers 

 being fet afide) if we make ^ = i5 and each of them equal to 

 unity ; the produds then with their refpedive fadors will be, 

 5f5 .^ — 3^ — 2 — x-V^'- y-x — 2 and a-' * — 3^+2 =x — ij- xx+ 2; 

 whofe difference never exceeds four in whatever manner x be 

 varied ; and the number of different fadors is alfo reduced to 

 four. 



That the number of different fadors of fuch produds as 

 are required of three dimcnfions, cannot be lefs than four, may 

 be thus demonflrated ; if there were but three different fadors 



one 



