22 



GENERAL THEOREMS, CHIEFLY PORISMS, 



PROP. 21. Problem. Fig. 20. To find the curve whose 

 tangent is always of the same magnitude. 



Analysis. Let MN be the curve 

 required, A B the given axis, s M a 

 tangent at any point M, and let a 

 ? be the given magnitude ; then, 

 SM.g. = SP . q . + PM . q . = a 2 ; 



77^ fi "i (1 3u 



or > # 2 + -^. '- = a *; and T-i = 



; therefore, dx = x V 2 V 2 - I n order to integrate 



y y 



dy 



this equation, divide V a 2 y* into its two parts, 



7 



y dy 

 and ; to find the integral of the former, 



a*dy 



a*dy a*dy 



therefore the integral of 



is a x hyp. log. 



part> ~ y y , 

 V a 8 - 2 



s 



a 2 y 2 ; therefore the integral of the aggregate - 



7 



a a _ i - a x h. 1. a 



or 



