532 MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 



pendiculars meeting the radius-vector and its continuation in E and F, then 

 AE = tan t, BF = cot t : also draw EG parallel to DA, FH parallel to DB, then 

 OG = sec t, OH = cse t. 



We shall afterwards reach the functions sine and cosine when treating of 

 recurring derivatives of the fourth order, and shall have the analogous construc- 

 tion shown in fig. 12. There OC is made equal to the function cos t, while 

 CD is made equal to the sin t ; but it will be shown of those functions that 

 cos t + sin f = 1, wherefore D must be a point in the circumference of a circle 

 described from ; AE is then the tangent, BF the cotangent ; and, to keep up 

 the analogy, if we draw EG parallel to DA, FH parallel to DB, OG becomes the 

 secant, OH the cosecant of t, t being represented by the double of the sector 

 OAD. 



20. If, as in fig. 2, the absciss OT be made proportional to the primary t, and 

 the ordinate TB to the function sus t, OA being the linear unit, the locus of the 



Fig. 2. 



point B may be shown to be the catenary thus. When OT increases by a minute 

 quantity dt, the ordinate TB increases by cat t.dt, and therefore the increment 

 of the arc AB must be \/(df + cat f . df) ; now 1 + cat f = sus f , wherefore 

 the increment of the arc is sus t.dt, which is just the increment of cat t, so that 



