G. W. Littlehales — Isolated Shoals in the Open Sea. 109 



The probability^? of having, at the same time, the error x 

 and the error y, or of deducing the geographical position P as 

 the position of the shoal, will be the product p t , p„ or 



(m—x) (n — y) 7 , . . 



p=- — J,^.' **•*» ( 3 ) 



an equation in which x can vary from zero to m, and y from 

 zero to n. It is, therefore, applicable to the first right angle 

 of the axes OX and OY, but, in order to make it applicable to 

 other quadrants, it is only necessary to change the signs of x 

 and y. 



Equation 3 then expresses the probability that A's determi- 

 nation of the geographical position of the shoal is in error by 

 x miles in longitude and y miles in latitude. 



If the center of the shoal were really located in the geo- 

 graphical position assigned to it by A, and B should succeed in 

 coming within r miles of it, he would find the shoal since its 

 radius is r miles. 



We have, therefore, as the second step in the solution of the 

 problem, to determine what is the probability that B will come 

 within a circular area, r miles in radius, having its center any- 

 where within the rectangle described about the true position 

 of the shoal with sides equal to the extreme errors to which 

 the determinations of latitude and longitude by A and B are 

 subject. 



To find the probability, P, of coming within any portion of 

 the rectangle of extreme errors inclosed by a curve whose equa- 

 tion is y~f{%), it is sufficient to integrate the expression (3) 

 between limits depending only upon y =f(x), and we shall 

 have, in the first right angle, 



= m i tfJ dx J( m ~ X ) **~" y ) dy ^ 



For a circular area of radius r, we shall have for the first 

 quadrant, 



p = mwJ ( m - x ) dx J ( n -y) d y 



and for the whole circle, 



v |/ r 2_ a .2 



p = n7rfj ( m ~ x ) dx J ( n ~y) d V 



or 



_ 2r 2 j n 2r 2r r 2 ) 

 mn ( 2 3m Sn 4?nn ) 



The probability that B will find the shoal depends upon the 

 concurrence of two independent conditions whose separate 



