Trigonometrical Surveying. 113 



a =ft -H {(sin ^ cosec &f -f (sin y cosec f»)* — 

 2 sin J cosec ^ sin y cosec ^ cosec (« — /3)}^ (!!•) 



Problem V. 

 To determine the same thing, if the given base have a paral- 

 lel instead of a perpendicular direction to the required dis- 

 tance. 



C 

 Let OL = a, LOK = «, LOC = y, CKL = ^, 



OKC zz ^^, OLK = K : V\ ^^^^^--^ 



CK=6, LK0=/3, LCO = ^, KCL = p, \\l 

 OCK = «, OLC = ^ 



By a similar method of investigation, 



hz=. a {(sin a cosec ^)- + (sin y cosec J)2 



2sin<« cosec ^ sin y cosec ^ cos (^ -j- ^)} 2 /12.) 



a-zzh'h' {(sin fit cosec /3)2-|- (sin y cosec 5)2 



28in« cosec/8 siny cosec 5 cos (^4- ^)}2 /i3.) 



Similarly, 



6 = « {(sin J6 cosec ^y -f- (sin g cosec 5)^ — 



2sin« cosec /8 sin ^ cosec 5 cos (4/ + «)} 2 (14.) 



a •=.h'^ {(sin» cosec /S)^ + (sing cosec 5) — 



2 sin* cosec i3 sing cose 5 cos (ij/ 4. *»)}2 (15.) 



There is another problem, of considerable utility in fixing the 

 position of a point with respect to, and its distance from, three 

 other given points. This is commonly known in this country un- 

 der the name of Townley's problem, though it appears to have oc- 

 curred to Snell a little earlier. It is given in most books on Tri- 

 gonometry. I first extended it to the method of finding the lati- 

 tude and longitude of a fourth point from the latitudes and longi- 

 tudes of three other places. In this form, it has been applied by 

 myself to fix the latitude and longitude of the Observatory of Edin- 

 burgh, when referred to three other points, whose latitudes and 

 longitudes are given in the Trigonometrical Survey. These are 

 given at full length in the second edition of my Tables, already re- 

 ferred to, and therefore it is unnecessary to insert them here. It 

 has since been used successfully by a naval officer in a similar man- 

 ner. 



VOL. XVI. NO. XXXI. JANUAEY 1834. H 



