Equations of the Second Order. 35 



Consequently by substitution, since p^p, 



cc + c' = cp Vl+pZ—py + clog ( */l+p*+p), 



which equation is identical with(/3), it having been already shown 



that jr = c. 



If the equation (/3) were integrated, an additional parameter 

 would be introduced; and from the foregoing reasoning it fol- 

 lows that, by giving different values to this parameter, c and d 

 remaining constant, the equations of all the involutes might be 

 obtained. But here an analytical circumstance must be men- 

 tioned the signification of which will require particular conside- 

 ration. The complete integral of the equation (ft) will contain 

 three arbitrary constants, and therefore cannot be the integral of 

 A=0, which is of the second order. It can, however, be shown 

 as follows that the differential equation resulting from the elimi- 

 nation of the three constants is verified by the equations A=0 



dk. 

 and — =0, the latter of which is a true analytical consequence 



(XX 



of the other. 



For by differentiating the equation (/3) to get rid of the arbi- 

 trary constant c', we obtain 



or 



= 1 +p* + yq -kq<Sl +p*, 



Differentiating the last equation to eliminate Jc, the result is 



x/l+i? 2 q* (l+p*f 



r being put for —-. If now the value of q be deduced from the 



equation A=0, and the value of r from -=— = (A=0 being 

 taken into account), it will be found that 



1 +P 2 a PQ 



<7= — , and r= ~-^ 



* V 1 + 



y L+P 



But these values of q andr, being substituted in the equation (7), 

 cause it to vanish. The inference to be drawn from this result 

 is, that the integral from which the equation (y) was deduced, 

 although it is incapable of directly satisfying the equation A = 0, 

 may still be regarded as a solution of that equation, inasmuch as 



D2 



