\ 



58 Solution of a Problem of Col. Silas Titus. [Jan. 



a = e±T.2'r^-i_ gj.^^ ^f ^jjg j3j|j quajj-ant (25) 



j__g±V-2'rV-i _ ^j.^^ qJ. ^jjg j^jjj quadrant (26) 



c = e±V-2'^l/-i=: area of the 21st quadrant (27) 



To prove that these are the true roots of equations (22), (23), 



(24), I substitute these roots for a, b, c ; then take the differential *, 



considering the sign ± as the differential sign. These substitutions 



give me 



/17 + 21\ 



g±V-2^V_l^e± V— 4— ;.2^v/-i ^ lgg±2^v- 1 (28) 



/IS + 21) 



g±\\2^^- l+g±V 4 ^27r^/_l -- I7e± 2^v'-l (29) 



By taking the differential, I obtain 

 ± (V. 2. y^^TT) e^ V-^-'v/— 1 ± (1I±11 2 , V-^:ri) 



/n + 2i\ , — 



e± \—^~) 2rV-x^ ±27rV'-l X 16e±27rv-l ^ 



the area of the 16th ring (31) 



/'l3 + 2l^ , — ~ 



e± (.— 4— ^27rV_l ^-t2T^/-lXlre±-'^^-^=^ 



area of the 17th ring (32) 



area of the 18th ring (33) 



The experimental quantities of equation (31) are reducible to 



(34) 



I i 19 . v^n ^ _ 1 I (Vid. equation 13) | 



{ 



(35) 



^± 17 ,r V j-^= _ 1 (36) 



e^ n'^ V-i=_i , (37) 



The equation (33) becomes 

 ^± 21 TT V^^ _ 1 (33) 



e* »5,r V - i3=_i (39) 



♦ This kind of differential is the true and strict meaning of Lemma II. Sect. IT. 

 Eook II. of Newton's Piincipia. (Momentum Genitte, &c.) The mannerin nliick 

 Newton has demonstrated this lemma entirely refutes every jioskiblc objection. 



