DIFFERENTIAL AND INTEGRAL CALCULUS. 95 



la and ^ increasing together -7^ is positive J , from which 

 the arc to any point may be found, 



COS ir COS 



/ sin ^r [ sin ^ 



= 2a /- - tani/r f- 



VCOS'T|T J COS 2 o/r 



sin - I - cos 2 



[ 

 - 



J 



(cOS 2 l|r 



therefore 



(/sin'xjr [ d^r \ /sin^ /TT alrX) 



25 = 2a \ ( f-r + -^ ] = 2a ^ + log tan - + -^ \ [ 



(\COS'^ JcOSi/ry Vcos'-^ V4 2/J 



+ (0=0) (s = when ^ = 0), 

 or 5 the arc measured from the vertex to any point is 



fsin-v/r /TT -\Jr\) 



a \ fr + loff tan - + ?- }[ . 



(cos'i|r \4 2/j 



Thus the arc from the vertex to the end of the latus rectum 



Again take the catenary 



c . x * du \ . * _* 

 JT- 5 (4 iff), ^ = -(s^- c) 



therefore e = tan^/r + sec>/r ; also s~c = seci|r 



c fir \|r\ 



and y = -- r . aj = c log tan h , 



cosijr' V4 2/ J 



c??/ c sini|r dv ds . c/5 



r ^ . _ L. K _ 5= gjjj ^ 



?^r COS a i|r ds d"^r Y d^r ' 



f ds c 



therefore -y = >2 , 



C?>|r COS i/r ' 



and 5 = c tantjr measured from the lowest point of the curve. 

 Next take the four cusped hypocycloid x^ -f y^ = a^; any 

 point on this curve may be represented by 



x = a cos 3 7 y = a sin 3 ; 



