ANALYSIS OF CURVES. 425 



Method of Analysing Curve (Summation). Plot one half wave 

 of the curve to a large scale on squared paper, the hori- 

 zontal scale being chosen to enable the curve to be 

 divided conveniently into a number of equal parts. These 

 parts may conveniently correspond to 30 of 6 each, 15 

 divisions of 12 each, or 12 divisions of 15 each, according 

 to the accuracy desired. 



In order to obtain the value of the fundamental curve 

 proceed as follows : Measure the height of the curve at 

 each of the horizontal divisions decided upon, tabulating 

 the values as shown in Table I., column 2, entering at 

 the same time the angle corresponding to each ordinate 

 in column 1. In column 3 enter the sines of the angles 

 given in column 1, as found in any set of mathematical 

 tables. Column 4 is then obtained by multiplying 

 together the figures in columns 2 and 3. 



Column 5 gives the cosines of the angle in column 1, 

 and column 6 is the product of these cosines by the 

 numbers in column 2. 



Column 7 is obtained by writing down the values of 

 the sines of 3 x angles in column 1. Multiplying these 

 sines by the figures in column 2 we obtain column 8. 

 Column 9 contains the values of the cosines of the angles 

 whose sines are in column 7, and the tenth column gives 

 the products obtained by multiplying them by column 2. 

 These columns of figures will give the fundamental and 

 the third harmonic. If the fifth or higher harmonics are 

 also present, further similar columns for sin 5 0, e sin 5 6 

 and cos 5 0, e cos 5 0, &c., must be added. 



After filling in the columns as described, add together 

 all the figures in column 4 and divide by half the number 

 of these figures, thus obtaining twice the mean value. 

 Carry out the same operation for columns 6, 8, 10, &c. 

 In this way the constants A 1} B lt A 3 , S 3 , <fec., are obtained. 



The expression for the curve is then given by substi- 

 tuting the values found for these constants in the equation 

 e = A l sin + B v cos + A 3 sin 3 + 7? 3 cos 3 + . . , . 



This equation may be put into a simpler form by 

 making the following substitutions : 



F l = A* + B* F a = ^A/ + Bi, &c. 



B ft 



tan 0! = - tan d> 3 = -. &c. 



