54 Colorado College Studies. 



enveloped by 3IN is of that class to which two aud only two 

 tangents can be drawn from a given point; that is, that it is a 

 conic. The limiting positions found in (8 ) are the points V, , V.,, 

 on the axis of symmetry PiPo at which tangents at right angles 

 to this line meet the curve. We may so far avail ourselves of 

 the analogy which the enveloped curve appears to bear to the 

 conic, as to name the point midway between T'^i and T', the 

 cenfrcy and the distance of the centre from either of them the 

 semi-axis. Then, if we denote by // the abscissa of the centre, 

 taken with the contrary sign, and by p the semi-axis, without 

 regard to sign, (since for the present we shall be concerned only 

 with the absolute value,) we shall have 



m' 4- ;//" , m' — m" 

 . aud p = -J ; 



•^ 2 



or, substituting from (8) the values of m' and m", 



^ -' {E^ - erf ^''^ ^' - (B^-cirf " ^^ 



We shall now prove analytically that the line MX does in 

 fact envelope a curve, viz., the cifclc, whose centre is at the 

 point — ^,0; and whose radius is jj. 



Thokem. — If a point move upon the circumference of a fixed 

 circle, and if two chords of this circle, which meet in this point 

 and hence move with it, continually touch a second fixed circle; 

 then the third chord which joins the extremities of the two 

 former, will also continually touch a fixed circle. Moreover, 

 the three circles have a common radical axis. 



When g aud _/:> represent any two quantities whatever, a 

 circle with radius j> and centre — g, has the equation 



ocr + / + 2gx + g- — ^r = 0; 



and the condition that this circle may be touched by any right 



line, 



Ix + ))n/ -\- n — 0, 



is found by eliminating one of the variables, and forming the 

 condition that the resulting equation may have equal roots. If 



