392 PRACTICAL MATHEMATICS 



Then if the ordinates y , y v y z , etc., are measured from the 

 fixed point along these radial lines respectively, 



2 

 A n = (sum of the vertical projections} 



2 



B n = (sum of the horizontal projections} 

 m l 



It should be noticed that the work will be made considerably 

 simpler if the number of ordinates is so chosen as to make the 



n_ 



angle a simple fraction of 27C. 

 m 



193. Considering a harmonic curve, the base line of which 

 extends from to 2?r for a complete period. Let the base line 

 be divided into 12 equal parts and the ordinates y Q , y v y z , . . . y u 

 drawn to the curve at the points of division. 



Then each division of the base will correspond to an angle - 

 or 30. 



1 



Then B = {y + y t + y z + ... y u } 



(1) Now A! = - y sin dQ, and B x = - 1 y cos dQ 



7T J o 7CJ o 



Hence A x = - (sum of the ordinates of the 0, y sin curve} 

 and B x = - {sum of the ordinates of the 0, y cos curve} 

 or A! = - {y sin + y 1 sin 30 + y z sin 60 + ... 2/ u sin 330} 



and B! = - {y cos + y cos 30 + 2/ 2 cos 60 + ... */ n cos 330 } 



If from the point O radial lines are drawn at intervals of 30 

 and the ordinates y , y lt etc., are measured along these lines. 

 (Fig. 128.) 



Then, resolving vertically, 



A i = <j ((2/1 - 2/7) sin 30 + (2/2 - 2/s) sin 60 + (2/3 - 2/9) 



+ (2/4 - 2/io) sin 60 + (?/ 5 - y u ) sin 30} 

 1 r 

 = Q 1(2/1 + 2/5 -2/7 -2/n) sm30 + (?/ 2 + j/ 4 -j/ 8 -2/ 10 ) sm 60 + j/ 3 -7/ 9 } 



and resolving horizontally. 



B i = g((2/o - 2/e) + (2/i - 2/7) cos 30 + (y 2 - y 8 ) cos 60 



- (2/4 - 2/io) cos 60 - (y, - / n ) cos 30} 

 = g(2/o ~ 2/6 + (2/i + 2/ii - 2/5 - 2/7) cos 30 

 + (2/2+2/io-2/4-2/ 8 ) cos 60} 



