194 On the RESOLUTION of 



Let AxB be any compound qi^ntity equal to another, 



CXD, and let ;« be any rational number aflumed at plea- 



fure ; it is manifeft that, taking equimultiples, AXwB =z CXw/D. 



If, therefore, we fuppofe, that A zz jiiD, it muft follow,^ that 



c 

 wB n C, or B zr — . Thus two equations of a lower dimen- 



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

 tion, we may aflfume the niultiples n and />, and form four 

 equations ftill more limple. By the repeated application of this 

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

 refolved into thofe of the firfl order, the number of which 

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

 the number of limple equations into which .a compound ex- 

 preffion 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 alfumed multiples. 



We fliall illuftrate the mode of applying our principle, in 

 the folution of fome of the more general and ufeful problems 

 conneded with this branch of analyfis. 



PROBLEM L 



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

 the fquares of Tvhich pall he a given number. 



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

 hypothefis, x'^ — y — ah ; but thefe compound quantities admit 

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



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

 bitrary, 



