On the Problem of the In-and-circumscribed Triangle. 267 



est distance of the pause-cone (2i inches), the place streaming 

 out is small and of inconsiderable densit)^ ; here, therefore, the 

 sphere also retains the density necessary for a spark. When, on 

 the contrary, the sphere stands within the pause-distance, the 

 extent of the surface streaming out, and its density are both 

 great, and the density of the sphere sinks immediately from its 

 highest value till it falls below that which is requisite for a spark. 

 It may be seen from this, that the sparks where pauses exhibit 

 themselves must be preceded by a streaming out, which termi- 

 nates in a sounding brush. In some positions of the cone this 

 streaming out may be detected by the eye, in others it is only 

 made manifest to the ear by a peculiar noise wliich accompanies 

 the sound of the sparks. The visible brushes which alternate 

 with sparks, appear near the beginning and the end of the pauses, 

 where the narrowest and longest sparks are observed. It occurs 

 here sometimes, that, at the commencement of the experiment 

 sparks are obtained, but during'^the continuance of it, none. To 

 obtain sparks again, in such a case, the experiment must be dis- 

 continued for a few minutes, or still better, the conductor must be 

 electrified negatively for a few seconds, and then the experiment . 

 continued. The absence of the sparks is here due to the elec- 

 trifying of the air, which alters the electric arrangement on the 

 pause-sphere, and which is neutralized when the air, by the ac- 

 cession of the opposite electricity, has become again non-electric. 



XL. On the Problem of the In-and-circumscribed Triangle. 

 By the Rev. George Salmon, Trinity College, Dublin*. 



^ I ^HE following is a direct investigation of the solution which I 

 -i- gave in the last Number of this Magazine of the problem, 

 " to find the locus of the vertex of a triangle two of whose angles 

 move on a conic V, and whose three sides touch a conic U." 



I first form the equation of the pair of lines drawn from any 

 point to touch U ; then the equation of either pair of hues join- 

 ing the points where these tangents meet V ; and lastly, form 

 the condition that one of those joining lines should touch U. 



I write the condition that \U + V shall represent right lines, 



and if V represents right lines, A' = (); and if, moreover, one of 

 these right lines touch U, we must have B2 = 4A©'. 



Now if P represent the polar of any point, the equation of the 

 pair of tangents drawn from that point to U is UU' — P^ — q, 

 To form the ecjuation of the pair of lines joining the points where 



* Communicated by the Author. 



