DISCONTINUOUS FUNCTIONS. 249 



These formulae will be found to be applicable 

 to the determination of any lines connected with 

 these discontinuous curves ; such for instance, to 

 find the tangent, radius of curvature, &c. &c. 

 without having recourse to infinite series, or to 

 their equivalents, definite integrals. 



It must be remembered, however, that the 

 connexion of the system will give us the following 

 relations amongst the functions, viz. : 



f(a) = f^{a)■, f,(b) = f,{b); f,(c) = {,{c), &C. &C (6) 



These equations will enable us to calculate the 

 two tangents which may be drawn to each of the 

 points C ; C, ; C2 ; &c. &c. ; as it is evident there 

 will be one tangent arising from the equation 

 1/ =. f(cc), and another from the equation fi(>r), 

 and this will obtain at all the other breaks of the 

 system. 



(11). If the relations f'(a) zz f/(a) ; f,'(6) - 

 f.i(b); fi'ic) zzfs'(c) &c. &c. exist, where f'(a) 

 &c. &c. are the diff'erential coefficients of f (.r) 

 &c. when .r =: a &c. then the tangents at the 

 points C ; &c. &c. will coincide. This relation 

 amongst the variable function of x which gene- 

 rates the discontinuous curve, will, I conceive, 



Kk 



