133 Gomometric Prollem.?. 



= tang.; therefore it becomes /, ASBr,ESD = if, CSD ; 

 agreeing with the equation before found for this case, p. 34. 



Prol'lem VIL* 



To find the relation between the angles ASC, BSE, ASB, 

 and CSE, when the point E is higher than C. (Fig. 15.) 



First we have AB = SB ^ ASB ; BD = SB^^BSD; 

 BE = SB f, BSE and AC = AS t, ASC = SB , ASB/, 

 ASC = BD = SB t, BSD, also DE = SB [t, BSE - 

 ^, BSD) ; hut DE* + DC« = EC" = SB' {t, BSE-^ BSD)' 

 + SB^ /^ ASB = SB^ (/, BSE - /, BSD]' + t\ ASB), and 

 CS = ASy, ASC = SBy, ASB/, ASC ; also SE = BS 



/,BSE; then by trig. c,CSE = ^^^1±I^L^^ = 



(SB/, ASB/, ASC)' + (BSY-, RSE) — SB- (/, liSE — /, BSDV + /',ASB) 



YsbT; As"^7i:sC TSB/ BSE 



/^ ASB f ASC + f, BSE - P, BSE + V.t, BSE *,BSD— i^BSD -i% ASB 

 = -^ : :^ . ,or 



^/, ASB/; ASC/, BSE 



/"-, ASB/'S ASC -!- ] + 2^ BSE t, BSD - t', BSD-r-, ASB 

 = c, CSK X 2/, ASB^, ASC/, BSE; but by Cor. to Pro- 

 blem V, ^BSD = /, ASB /, ASC; consequently /SASB 

 / % ASC + 2 /, BSE f, ASB t, ASC - /\ ASB /% ASC - 

 'p, ASB = 0, CSE X 2 /, ASB/, ASC/, BSE - 1 ; but /* 



— P = rad.^; whence by reducing and dividing the whole 

 equation by 2, we get c, CSE f, ASC /", HSE - c, ASB = 

 t, BSE/, ASC; wh'ich divided bv /, ASC/, bSE, becomes 

 c, CSE - c, AS > c, ASC c, BSE = /, BSE/, ASC c, BSEc, 

 ASC, and since /, x r, = s, therefore it becomes c, CSE 



— c, AS3 c, ASC c, BSE = s, BSE i, ASC, for the required 



* This problem may thus be solved by spherics : 



Let Z (tig. 16.) be the zenith of the observer at the station S, then the arc 

 ZD will he the complemeat of the angle CSA of the object AC, and ZF the 

 complement of the angle of elevaiio.i ESB of the object EB taken at the 

 point S ; but bv what is shown, p .33, the angle DZF = the angle ASB. Now 

 in the triangle DZF v.-e have jj^iven by observation ihe three sides to find the 

 angle DZF, which may be had by eiih:r of the following formula. (Cagnoli, 

 4(ia,464.) 



.,.DZF=^/l-i:AzL^5)i^('lz_E!) 

 s, ZD s, ZF 



./s, 1 S s, r-t S - DF) 



s, ZD .V, ZF 



where S is put to represent the sum of the sides. 



relation. 



