ii2 A NEW SOLUTION 



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



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



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 ofr~ tan <p, and 



z — tan - is thus exprefTed r zz ? *~* \ '• and r becomes infi- 

 nitely great ; ift, When i — 32* no; 2dly, When z is infi- 

 nitely great. 



Now the values of z, derived from the equation i — 32* zz o, 



are z zz + y- 9 and z zz — -r- : And thefe are precifely the 

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

 Again, we have r ~ zz \~-Z* '- And it is manifeft that the 



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

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

 I zz I and z zz 3r. But r zz ~ M ^ ~ J therefore, 



,M« -f- 3N£ 

 * = M + N and * = H^~. ^#1^ ' If n ° W WG 



write — for b, in this exprefiion, the infinite quantity a may be 



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

 in finite quantities only. The exprefiion being properly redu- 



ced, we fhall have, * = M ( 3 J- „) + *N« ' 



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

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



n 



pute the third root, x zz 2N „^ M(3 «-«y 



Applying 



