28 



On Equations of Condition for a Quadrilateral, common or re-entrant. 

 Captain K. Shortrede, Assistant Surveyor General. 



By 



Some years ago when engaged in investigating a general method for 

 disposing fairly of the errors which occur in observing angles, the fol- 

 lowing properties of the angles of a quadrilateral occurred to me as 

 furnishing equations of condition proper to be used in such cases. Some 

 of the properties were new to me at the time, but it is possible enough, 

 most of them may have been known long ago to others. 



In the quadrilateral A B C D with its diago- 

 nals, for the sake of conciseness and uniformi- 

 ty, let A denote the angle B A C, A2 the an- 

 gle CAD, and A3 the angle B A C ; and 

 similarly for the other angles, those on the left 

 hand side having 1 , as their distinctive mark ; 

 those on the right hand having 2, the whole 

 angle of the figure being distinguished by 3. 



When the quadrilateral becomes re-entrant as 

 in the 2d figure, a similar notation is used as in 

 the margin. 



There are two classes of these equations of 

 condition, of which, one, depending entirely 

 on the summation of the angles, may be term- 

 ed the class of angular equations : the other 

 depending on the products of the sines of 

 the angles taken in a certain order may 

 be termed sinal* equations. 



In the first figure we have the following angular equation : — 



Ai -{- As = A3 } These may be termedf Totopartial equations. 

 , * x Bi -f- B2 = B3 I r f These are not peculiar to figure or to space, but 

 Ci + Cs = C3 1 i are expressions for a universal property, common to 

 Di + D2 = D3 ) l a n sorts f quantity. 



• This word I prefer to Sinical, which, though formerly used, is now nearly obsolete : it has the 

 advantage of being shorter, besides, not being liable to be mistaken for another word of similar sound, 

 but of totally different meaning. Moreover the word sine being of apocryphal origin, we need not, 

 for the sake of a fancied analogy, bind ourselves to the use of an inconvenient term 



