384 Mr Murphy on the Inverse Method of 



put 1, 2, 3 &c. successively for n in this expression, and we shall 

 generate the series for fAy), namely 



1 ! , ! , 1 a 

 I+i + - + - + &cl 



1 1 1 



-3~5-7 + &C - 



1, 



a result which may easily be verified, by geometrical considerations. 



24. We shall terminate this Section with an example, in 

 which the function is, infinitely, discontinuous. 



Fig. (II). An infinite series of equal and similar arcs of any 

 kind, are ranged at equal distances, across the horizontal right 

 line Ax, the successive arcs being included between pairs of 

 equidistant ordinates ; to find the equation of the system. 



(Note, the lower branch of Fig. II, only differs from the upper 

 in supposing the extreme points («„, ft,) («,,//,) &c. to coincide). 



Let y=f{x) be the equation of the curve of which a„cj>„ is 

 an arc, x being the abscissa for the curve when continuous, and a 

 the abscissa for the discontinuous curve; let the distance c„c, = 1. 



When « is between - \ and \ ; y-f{ a ) is the equation to a„cj> . 



\ and | ; y = f{a-l) «iCi*„ 



| and | ; y = f( a - 2) a.cj),. 



Similarly, 

 when a is between - 5 and -|; y=f{a + \) a_iC_,Z»_! 



&c. &c. &c. 



