194 On the RESOLUTION of 



Let AxB be any compound quantity equal to another, 

 CXD, and let m be any rational number affumed at plea- 

 Aire ; it is manifeft that, taking equimultiples, AXwB — CXwD. 

 If, therefore, we fuppofe, that A z= niD, it muft follow, that 



7/1B — C, or B rr — . Thus two equations of a lower dimen- 



fion are obtained. If thefe be capable of farther decompofi- 

 tion, we may affume the multiples n and p, and form four 

 equations (till more fimple. By the repeated application of this 

 principle, an higher equation, if it admit of divifors, will be 

 refolved into thofe of the fir ft order, the number of which 

 will be one greater than that of the multiples aflumed. Hence 

 the number of fimple equations into which .a comppund ex- 

 preflion can be refolved, is equal to the fum of the exponents 

 of the unknown quantities in the higheft term. Wherefore a 

 problem can be folved by the application of this principle, only 

 when the aggregate fum, formed by the addition of the expo- 

 nents in the higheft terms of the feveral equations propofed, is 

 at leajl equal to the number of the unknown quantities, toge- 

 ther with that of the aftumed multiples. 



We fhall illuftrate the mode of applying our principle, in 

 the folution of fome of the more general and ufeful problems 

 connected with this branch of analyfis. 



PROBLEM I. 



Let it be required to find two rational numbers, the difference of 

 the fquares of which fhall be a given number. 



Let the given number be the product of a and b ; then by 

 hypothecs, x 2 — y 1 == ab ; but thefe compound quantities admit 

 of an eafy refolution, for (x-\-y) (x—y) zr aXb. If therefore we 



fuppofe x-\-y — ma, we fhall obtain x — y — — ; where m is ar- 



m 



bitrary, 



