ORIGINATING THE CONSTRUCTION OF A HYDROSTATIC QUADRANT. 187 



and moreover, it has been further proved, 'that 

 5 =r vers.a; vers.(< a;) ; 



therefore, multiplying both sides by, we shall get 

 m {vers.a? vers.(< x}} ; 



S 5 



let these two values of be compared with one another, and we 

 shall obtain 



vers.(20 a?) vers.(0 #) m {vers.z vers.(</> x)} ; 



or multiplying by s, we have 



s vers.(2^> ar) s vers.(^> x} zz: s' vers.a? s' vers.(0 x). 

 Now, by substituting for the several versed sines, their values in 

 terms of the cosines and radius, we shall obtain 



s{cos.(0 x} cos. (2^ a:)} s'{cos.(0 x) cos. a- } 



from which, according to the arithmetic of sines, we get 

 s {cos.0 cos.a: -f- sin.0 sin.a; cos.2^ cos. x sin.20 sin. a;} zz: s' {cos.< 



cos. a; -j- sin.0 sin. a? cos. a;}. 



Let all the terms of this equation be divided by cos. a;, and it 

 becomes transformed into 

 s cos.0 -{- s sin.0 tan. a; s cos.20 s sin.2^> tan. a: zz: s r cos.0 -f~ s/ X 



sin. ^ tan. x 5'; 



therefore, by bringing to one side, all the terms that involve tan. a:, 

 we shall have * 



sin.^(s' s)tan.a; 5 sin. 20 tan. x n: scos.2^ -|- (*' s)cos.^> s' ; 

 hence, by division, we shall obtain 



s cos.2<6 4- (s' s) cos.0 5' 



tan x "~~ - 



s sin. 20 + ( s> s ) sin.0 (142). 



If the equation which we have just obtained, be compared with that 

 numbered (141), it will readily appear, that the one might have been 

 deduced immediately from the other, by simply substituting s' for m, 

 and s for unity, in the several terms of the numerator and denominator; 

 but in order to render the formation of the formula more intelligible, 

 we have thought proper to trace the steps throughout. 



200. The practical rule for reducing the above equation, will require 

 a different mode of expression from that which we have given in the 

 rule to equation (141), but it will not be more operose ; the rule is 

 as follows. 



