G. W. Littlehctles — Isolated Shoals in the Open Sea. 107 



likewise, the longitude from the true longitude. If we call the 

 differences between the true latitude and each deduced latitude 

 errors of latitude and lay them oft", according to their signs, to 

 the right and left of an assumed origin, and then, correspond- 

 ing to each error as an abscissa, erect an ordinate of a length 

 proportional to the probability of that error, these ordinates 

 and abscissas will be the coordinates of the probability curve. 

 And, likewise, if the errors in longitude were found and 

 plotted in conjunction with their probabilities, a similar curve 

 would be developed. 



In this investigation the probability curve, ordinarily repre- 

 sented by Laplace's formula, y = ce~ aH " will be replaced by 

 two equally inclined straight lines AB and AB' as shown in 

 figure 1. 



1. 



This substitution, which has been employed by Helie in his 

 Traite de Balistique Experimentale and referred to by Wright 

 in his work on the Adjustment of Observations, causes an 

 appreciable but extremely small error which has no practical 

 significance when we consider that, from the nature of the 

 calculations about to be made, absolute precision is not to be 

 sought. 



The probability of having an error between Oc = x (figure 1) 

 and x+Ax, to the right of the axis OS, is equal to s . dx. As, 

 in this case, OB and OB' measure the extreme errors, all pos- 

 sible errors are comprised between zero and OB, and zero and 

 OB' ; and the sum of all the elements which are singly rep- 

 resented by «? . dx, or the area of the triangle ABB', should be 

 equal to unity, which is the measure of certainty. The equa- 

 tion to the straight line AB will be, calling m the extreme error 

 OB and b the intercept on the axis of S, 



s x 



T + -= 1 

 o m 



but, since the area ABB' = bxm = 1 or b = — this equation 

 becomes 



1 

 m 



