500 Mr. JR. H. M. Bosanquet on the Beats 



tion in question so far as it is connected with the subject of 

 the paper. 



79. The curves arc referred to an axis of x, along which the 

 wave-lengths arc measured, and an axis of y parallel to which 

 the displacements are measured. X and X' are the wave- 

 lengths on the paper of the two primary curves. If it is 

 required to consider a question of frequency, the paper must 

 be supposed to be drawn past the observer with velocity v, 



when the frequencies will be -, —, respectively. 



A- A. 



80. The tangent of the inclination of a curve to the ^-axis 

 will be spoken of shortly as the u slope." 



It is assumed that ^V = ^X + 8, where p, q are integers, 

 q >p, and S is small. 



81. The equation of the resultant of two primary curves 

 may then be written 



y = E cos27r!- + Fcos Tr (^'— a). 



A. A. 



The slopes of the two single curves are 



2-77- -j-, . X 2lT -p, . 2-7T . v 



A AAA 



The ratio of the coefficients is 



X'E pE 



— = _ nearly. 



When this ratio is much greater than unity, the resultant 

 slope is nearly that of the first term. When it is much less 

 than unity, the resultant slope is nearly that of the last term. 



82. The general expression for the resultant slope is given by 



dy 2-7T ^ . X 27T -^ . 27T . N 



-f = — E sin 27r - -ryFsinrr (x—a). 



ax A AX X' v J 



The vertices of the resultant curve are obtained by equating 



-~ to zero, whence 



E . 2ttx F . 2tt 



— sm-jj- + -,sm-^-(x — «) = 0. 



83. Case I., where F is great, and the first term negligible 

 compared with the second (r-^- small). 



Here the vertices are those of the second component of the 

 curve. Consequently, in every cycle of p and q vibrations of 

 X and X' respectively, the q vertices of X' appear, those of X 



