312 Mr. H. H. Turner. On the Correction to the [Apr. 1, 



We have 



j|cos 3 Xi\cos2Z^=^j ^sinX + sinSxJ^cos 2ldl 

 1 f* 



= l2j i 9sin (^ + a7 )+ sin3 ( Z + aj )} cos2Z ^ 

 + j*^(9 sin c + sin 3c) cos 2 1 dl 



+ J* - { 9 sin |(Z + + sin f(Z + y) } cos 2Z dl. 



We may thus simply travel round the boundary omitti ig the 

 places where X,= constant : being careful to go round all the pieces of 

 land in the same direction. If we suppose Z=<* to be the meridian of 

 Greenwich, and the land to be in the northern hemisphere, the 

 direction indicated above is the wrong one for obtaining the value of 

 the integrals over the land, for the longitudes increase to the left ; 

 but by following this direction we shall obtain the values over the 

 sea as is in reality required. 



The result of integration has, of course, a different form for each 

 form of the relation between I and X representing the boundary. In 

 computing the numerical values of the integrals, it is convenient to 

 consider together all the parts of the boundary represented by similar 

 equations. 



Below are given as representative the forms which the numerator 

 of the first integral % assumes for different forms of the boundary, 

 the quantities within square brackets being taken within limits. 



Form. Value of Integral. 



+ \=Z + a; +JL_[i C os (5Z + 3z) + 3 cos (3Z+a?) + cos (Z + 3.r) 



— 9cos(— l+xj] 



\=x .... +-2 1 ¥ (9sinaj + sin 3a;)[sin2Z] 



l=x .... Zero 



\-2l-\-x — ^[f cos (M+x)— 9Zsin#+£cos (8Z+3.?) 



+ ^cos (4Z + 3a>).] 



X=U+x — ^[f cos(6Z+a5)+-|cos(2Z+aj)+ 1 i r cos(14Z + 3aj) 



+^cos (10Z + 3.U)] 



+ 2\= l + x ± ^[iiL cos i(5Z + x) - 6 cos J ( - 3Z + x) 



+ f cos Jt7Z + 3a;)-2sinK-Z + 3.73)] 



-3\=Z+.tj .... + T V[-y- cos K7Z + *) ~nr cos K-5Z + 2) 

 +^cos (3Z -f- — cos (— l + x)~\ 



