ON THK TRIGONOMETRY OP THE PARABOLA. 81 



Now sec* + tan$=(sec^+tan0)", as shown in (6). Hence 



n(m . *)— wn(m . <p)=m[sei'. 4> tan ^—71 sec <{> tan <^]. 

 Let seC(j) + ta.T\<p=\, then sec4> + tan 4>=X'', and 



sec^=^±^\tan^=^i=^. 

 We have also sec *= — !- , tan 4>= . Hence 



U(m . <P)-nTl(m . ^)=m |-(^^"-X-='")-^(X^ - X-^)!. . . (23) 



3 5 



-, sec^=-, X=2. Then 



n(m . *)-3n(m . .^)= ^ /'liiV. 



3 5 



Let ^^=3, tan (6=-, seC(i)=-, X=2. Then 



When w=4, 



n(m . $) -4n(m . ^)= »2 ^ • ^^ • ^^"^ ■ 



210 



and so may n be taken any other integral number. 



XXI. The equation (20) affords a very simple mode of expressing the real 

 root of a cubic equation. 



Let the cubic equation under the ordinary form be a^+px-=-q. 



Let the parabolic equation tan^a)+ — tan u=— — be written 



4. 4, 



hence 



Sw3 Wi^ 



tan^w+ tan w= — tan £2, 



4 4 



3 2 in 2. r^ 

 p= -mr, g= — tan Q.. 



4 4 



Now since the value of x found by the ordinary methods is 



we shall have 



2a;=m \/sec£2+ tanii— »2 -v^secil— tanii, . . . (24<) 

 and 



-=Vf' — I.VI- 



When the sign of jo is negative, the solution must be sought in the trigo- 

 nometry of the circle. 



Section HL On the Geometrical Origin of Logarithms. 

 XXIL In the trigonometry of the circle we find the formula 



cv— f„„a tan^^ tan^^ tan^S , „ , . 



d— tan ^— —_ + ___ ;^-+&c (a) 



1856. ' G 



