

AN INVESTIGATION INTO THE EFFECTS OF ERRORS IN SURVEYING. 865 



That is to say, the theoretically perfect triangle for triangulation purposes is an 

 isosceles one, having the angles at the base 56° 15' and that at the apex 67° 30'. 



If a triangulation could be laid out composed entirely of triangles of this perfect 

 shape, the lengths of the sides would gradually decrease as one proceeded further and 

 further from the base. Now, this would not be desirable, since, as the sides shorten, 

 centring errors would in time cease to have a negligible influence, and the ultimate 

 effect would thus be to increase the average error in the angles. Again, with a scheme 

 in which the triangles were gradually being reduced in size, more triangles would be 

 needed to cover a given area of ground. Therefore, although the result last deduced 

 possesses a considerable theoretical interest, there is no reason to depart from the 

 opinion generally held, to the effect that the best shape of triangle for practical 

 purposes is the equilateral. 



Attempts are sometimes made to prove the equilateral to be the best-conditioned 

 triangle by a graphical method ; the premises, however, are generally defective, and in 

 some cases quite unreal. The writer believes the problem to be too intricate to be 

 treated adequately by any graphical -mode. 



The curves, fig. 5, are constructed from equation (25) ; they show the relation 



between the average fractional error, — , in the side, a, of an isosceles triangle, ABC, in 

 which the apical angle, C, assumes all values between 0° and 180". For curve A the 

 fractional error in the base, namely — , is assumed negligible ; for curve B it is taken as 



1-^20,000, or 5 x 10" 5 , and for curve C as 14*5000, or 2 x 10~ 4 . 



In all three cases the average error, v, in angle, is taken as 10 seconds (4 '8 5 x 10 -5 

 radians). 



These curves supply a considerable amount of information of practical importance. 

 They provide, first of all, a further justification in taking the equilateral as the perfect 



a 



shape, the difference in — being inconsiderable as between (7=67° 30' and C=60°. 



Secondly, curve B, which is constructed with — and v of similar magnitudes — a case 

 pretty common in practice — illustrates how rapidly — increases when C exceeds 120° 



CO 



or becomes less than 30°, but also shows that the rate of increase is not so rapid when 

 C assumes unduly large as when it assumes unduly small values ; it may therefore be 

 taken to demonstrate the well-known rule to the effect that no important triangle should 

 have an angle less than 30° or greater than 120°. On the other hand, curve C shows that, 

 when the error in base greatly exceeds that in angle, these limits may be set much further 

 apart without any appreciable reduction in accuracy. Indeed, when v=12 seconds, 



and — = 1-^5000 the angle C may be given any value from 15° to 145° without risk. 



We learn, then, that whether a triangle is to be considered permissible for triangulation 

 TRANS. ROY. SOC. EDIN., VOL. XLVII. PART IV. (NO. 28). 127 



