TRANSACTIONS OF SECTION A. 459 



and, therefore, the intrinsic equation of the evolute is 



sin< ^ > = #(/ / ^ ) 8in ^)- 



6. If the tangential equation of a curve he given, we find the tangential equa- 

 tion of its evolute by equations (6) and (9). For if j>=/($) he the tangential 

 equation, the intrinsic equation of the evolute is 



s = 2/'(tf>) cos <£+/"«>) sintf; 

 let this be denoted by F($) and by art (4), the tangential equation is given by 

 equation (6). 



Now from the value of F(<f>) we get 



F'(0) sin = 3/"(<£) sin cos $ +f'"{<t>) sin 2 -2/ / (<£) sin 2 $. 

 Hence we get 



/ cosec 2 <£ { /l"(0) sin <f>d<j>} cl<f> =/'(<£) +/(<£) cot <f> 

 .'. "=/'(<£) +/(<« cot 4>, 



OT v = Mf {(t>) Sin< ^)' (10) 



sin 

 is the tangential equation of the evolute. 



Hence it follows that if v Xi v 2 , v 3 , &c, denote for the successive evolutes what v 

 denotes for the curve itself, that 



"iSinc^^Vsin^j 



^sin0 = ^(Vsin0j, 

 and in general 



,„sin^ = ( < | ) )»(Vsin0). C 11 ) 



7. From the preceding article it is easily seen that we have for the w th involute 



tr^M^'\ m ^)' (12) 



or if we interpret symbols of differentiation with negative indices as denoting inte- 

 gration, the formula (11) includes both evolutes and involutes, according as n is 

 regarded positive or negative. 



8. From the tangential equation of a curve we get the intrinsic equation of the 

 involute. Thus let v = F ((£) be the tangential equation of the involute, then we 

 have from equation (4) 



S-H^ ■""♦)■ 



d(p d<j> 



sin <j) 

 and from equation (12) we have 



■A 



F(<£) =//(<£) sin <£,/<£ 



or, as it may be written, 



sin cf) 

 . f | = ^(/ W sin0) + /(/(0)sin0 # ) 



.•. «=/(<£) sin + / / (/($) sin <f> dcfAdcp, 



S= { 1 + (4) '}W^*- (13) 



