OF ARTS AND SCIENCES : FEBRUARY 8, 1870. 205 



or, rejecting the useless factor \ — a, 



X" 3 + a b 2 = 0, 

 whence 



x" = - vTb 2 , 



and by interchanging a and b, 



y" = - f^. 



And thus the length of the normal 



y/ tf - a) 2 + (/ - b) 2 = [(a + ^Tb 2 ) 2 + (b + ^b) 2 ] 1 



= [a3 -j- baj2. 



Consequently, 



if a 3 _j_ b? <^ 1, there will be four real roots ; 



" a f -4_ bf = 1, there will be four, and two will be equal ; 



« a3 _|_ bs ^> 1, there will be only two real roots. 



We will now show how to arrive at a direct solution of the problem 

 by the employment of trigonometric formulas. If tan <r is taken for 

 the unknown quantity, the equation, on which the solution of the prob- 

 lem depends, is 



[c cos /3 tan <r -\- c sin /3] 2 (1 -f" ta n 2 o-) = tan 2 o-, 



or if we put tan a- = x> 



(x + tan^( x 2 +l)=^, 

 or, expanded, 



x < + 2 tan /3. x 3 + ^=^ x 2 + 2 tan /3. x + tan 2 /3 = 0. 



A quantity p may be assumed, such that this biquadratic shall be 

 resolved into the two quadratics 



„ . _ sin a cos (/3 -4- w) , a , A 



y -+- 2 — „ y -4- tan |8 tan u = 0, 



x ~ cos cos 2 /i A ' M r 



„ _ cos u sin (/3 — u) . _ - . A 



y 2 4- 2 — — y -4- tan /3 cot u = 0. 



x ~ cos /3 cos 2 /* * ' M ^ 



That this is possible will be evident on multiplying the left-hand 

 members of these equations together, for after some reductions easy to 

 make, all the coefficients, with the exception of that of y 2 , will be found 



