260 EXAMINE. 



Fro; ion (7) we may find scc^ in terms of y, 



i os ^ and sint/r can also be found ; thus by substituting 

 in (5) we could obtain the equation between x and y: this 

 equation however would be very complex. 



In a particular case we may easily obtain nn ap]n <ximate 

 value of y in terms of x. Let X = ?/. u (5) may be 



written 



1+sinrir 

 , 



COSifr 





r 



fore ^- = 6 e 



1 + Sill i|r 



therefore by addition and reduction 



therefore tan> = J ( 



c 

 Now suppose - is a very s 



* jt 

 and v for \ (e 4- e c ) ; then the last equation gives 



c * 



Now suppose - is a very small quantity, put u for \ (e e e<) 



from this we can find tan -ty approximately, and then 



will be known approximately, and by substituting in (7) we 



shall obtain approximately y in terms of x. 



Equation (2) may also be written 



therefore -^ (meg ^ = m'g ; 



therefore, by integration, 



