226 PEOCEEDINGS OF THE AMERICAN ACADEMY 



having a number of points we wish to see if they lie on any curve 

 of the second degree. For instance, suppose the polar co-ordiuates 

 of the various points given, with the pole at the focus : then r = 



:; , in which we wish to see if any values of m and n satisfy all 



1 + w cos V J J 



\ \ n 

 the conditions. The equation may be written — = ! cos v. which 



becomes linear if 7 = J^ and cos v = X. In the same way, if referred 



to its centre, --, -!-■-,= 1, make X= x^ and Y= w-, when —, and -:; 



are obtained. 



5. Periodic Functions. In the study of periodic functions the equa- 

 tion y =z m sin (« x-\- p) is assumed, in which m determines the maxi- 

 mum amplitude, n the period, and p the phase. If m is given = a, 



n and p may be found by writing nx -\-p)^=^ sin"'-' ( ) ' ^^^ using as co- 

 ordinates X and sin~^ ( ) • ^^ ^^ period is given, or n = h, we have 

 y = ni sin b x cos p -\- m cos b x sin p, or, dividing by cos p, — '-^-r- 

 = m cos p tang bx -\- m sin p, in which we may make — ^y- = Y, 



tang bx ^= X, and thus determine 7n' = in cos p and 71' = m sin p. 

 From these two equations we finally obtain tang p = —^ and m = 



6. Lissajous' Carves. The wonderful variety of curves obtained by 

 Lissajous, b}' mirrors attached to tuning-forks, may all be reduced to 

 straight lines by this method. They may all be represented by the 

 equations x = sin v and y = sin (mv-\- n), in wliich m represents 

 the interval of the forks, and n the difference of phase. Eliminating v, 

 we have m sin~^ x -\- 7i = sin~^ y, which at once takes the linear form 

 when the co-ordinates X = sin~^ x and T'^sin"-'^^ are employed. 

 To test this, a curve was drawn with an instrument devised by the 

 writer (Journ. Frank. Inst., Jan. 1869), and forty-eight points on it 

 measured, corresponding to variations of v of 30°. In the following 

 table a portion only of the results are given. The first column gives 

 various values of v, the second the measured value of the arc whose 

 sine is y, or of in v -|- n. Constructing a curve with these co-ordinates, 

 we obtain very nearly a straight line, with equation y = ||^ v -\- 4°, 

 from which we infer that the difference in phase of the two forks at 0° 

 was 4°, and their interval a little more than a fourth (3 : 4), 4* = .755. 

 Tlie differences, as given in the fourth column, are very small, con- 

 sidering the roughness of the measurements. 



