112 A NEW SOLUTION 



And fince n and b are infinitely great, it is manifeft that thefe 



two roots are equal to one another, and each — | zi -• 



We can infer nothing with regard to the value of the third 

 root, derived from the infinite tangent, unlefs we can afcertain 

 the proportion which that infinite tangent bears to the infinitely 

 great quantities a and b. The general relation of t =z tan (p, and 



■z zz tan - is thus exprefled r zz -^^ ^^ : and r becomes infi- 



3 1 — 3« 



nitely great; ifl. When i — 32= = 0; 2dly, When z is infir- 

 nitely great. 



Now the values of z, derived from the equation 1 — 32^ =: o, 



are 2 :r + -7-, and z z=. — -r- : And thefe are precifely the 



values of 2 ufed above in determining the two equal roots. 



Again, we have - — ^ ^^\ : And it is manifeft that the 



Z I jK 



greater z is, the nearer -approaches to - : fo that, vikimately, 

 when r and z are greater than any finite magnitudes, we have 



1 = J and 2 = ;iT. But r z= "j^ ^^ ; therefore, 



2 - M 4- N ^"^ '^ ~J^^=- ' ^Ua+,^b ' If now we 



* ~ AJ _j- isi 



write — for h, in this exprefCon, the infinite quantity a may be 



thrown out by divifion, and we fliall have the value of the root 

 in finite quantities only. The expreffion being properly redu- 



- n It 1 2Mot + N(3» — ni) 



ced, we fliall have, x = ^173^^13^+ oNT/ 



When, therefore, Qj= o, two roots are equal to one another, 

 and each zz - : And we have this formula by which to com- 



putc the third root, x - ,t^„ JTmcj^;-;,) - 



Applying 



