tion ; required, considering that every observation is liable to error, in 

 what way these quantities may be most accurately determined. 



RULE. Substitute the quantities known by observation fory and x t in 

 the given formula (each observation being supposed to afford a value both 

 of # aud#), and thus, as many equations of condition will be obtained, 

 as there are observations. If these exceed the number of quantities to 

 be found, or of the equations wanted, let there be composed from the ad- 

 dition of them into separate sums, as many equations as are necessary, 

 each consisting of as many of the given equations as the question admits 

 of. From the equations thus obtained, the quantities sought may be de- 

 termined with the least probability of error. 



Suppose the general formula to be 



y = A sin. x 4. B sin. 2 x, 

 and that from observation we have eight values of x and y t viz. 



Values of 

 140 

 135 

 130 

 125 

 120 

 115 

 110 

 105 



Values of y. 

 73'.5 



80.2 



87.0 



94.1 



99.5 

 104.5 

 107.5 

 110.2 



Hence, 



.6428 A .9848 B = 73.5 



.7071 A 1.0000 B - 80.2 



.7660 A .9848 B = 87.0 



.8191 A .9337 B 94.1 



.8660 A .8660 B = 99.5 



.9063 A .7660 B 104.5 



.9397 A .6428 B =r 107.5 



.9660 A .5000 B = 110.2 



By adding the first four into one, and also the second four, we get 

 2.9350 A 3.9033 B = 334.8, and 

 3.6780 A 2.7748 B = 421.7 j 

 and therefore, 



- 1.7 ~ 2.7748 X 334.8 



3.678 X 3.9033 2.935 X 2.7748"' 



or A = i.55*00. 

 103 F3 



