298 Prof. G. H. Darwin. [June 19, 



sine of any angle lying within a given 5 of the circumference as 

 equal to the cosine or sine of the middle of that 5. 



The process then consists in the grouping of the heights according 

 to the values of their Vs (V m , V s , ^V m , as the case may be). The 

 heights in each group are then summed. Since the L.W. heights are 

 all negative, they are treated in a separate table, and are considered 

 as positive until their combination with the H.W. at a later stage. 

 We shall, for the present, only speak of one of these groupings, 

 taking it as a type of both. 



Since C - >S (a + 180) = , the eighteen groups forming the 



3 rd quadrant may be thrown in with the 1 st quadrant by a mere 

 change of sign ; and the like is true of the 4 th and 2 nd quadrants. 



Since cos (180 a) = cos a and sin (180 ) = sin , it follows 

 that we have to go through the 2 nd quadrant in reversed order, in 

 order to fall in with the succession which holds in the 1 st quadrant, 

 and, moreover, the cosine changes its sign, whilst the sine does not 

 do so. Hence the following schemes will give us the eighteen groups 

 which all have the same cosines and sines : 

 for cosines 



(lst_ 3 nr)_(2n<i-4 tb ) reversed, 



for sines 



(ist_ 3 rd-) + (2 nd -4 th ) reversed. 



Thus, one grouping of the heights serves for both cosines and sines, 

 and, save for the last step, the additions are the same. 



The combination of the H.W. and L.W. results is best made at the 

 stage where 1 st 3 rd and 2 nd 4 th have been formed. 



The negative signs for the L.W. results are introduced before 

 addition to the H.W. results, and total 1 st 3 rd and 2 nd 4 th are thus 

 formed. 



After the eighteen cosine and sine total numbers are thus formed, 

 they are to be multiplied by the cosines or sines of 2 30', 7 30', 12 30', 



.... 87 30'. The products are then summed so as to give 27i C ^ 



Sill 



It was noted at the beginning of this section that we also have 



sums of the form 2 . These sums are obviously made by entering 

 sin 



unity in place of each height, and, of course, not treating the L.W. as 

 negative. Thus, where the H.W. and L.W. are combined it is not neces- 

 sary to change the sign of the L.W., as was done in the combination of 



H.W. and L.W. for ~S,h These summations are considerably less 

 sin 1 



laborious than the others. 



In the case of the tides M 2 and S 2 , the division of the sums 2/t . 



