M. B. PORTER—SPHERICS. 
53 
In figure 12. Let L, L', L", Id" be four positions 
of the intersection of the legs tracing the conic A. 
LL'SS'a B.LL'SS'7\ A.PPT"P" , 7\B.PPT"P w . So 
that A, B, PP'P"P W are on a conic. 
Fig . 12. 
CONDITIONS NECESSARY TO DETERMINE A CONIC. 
22. 1 ° . Five lines or five tangents [11]. 
2°. The focus and three tangents, or the cyclic line and three 
points [5], 
3 ° . The focus, the directrix, and one point; or a cyclic line, its pole 
and one tangent (another point could be determined as the harmonic 
conjugate and thus both cyclic line would be determined (22-4°) ). 
4° . The focus, the directrix, and one tangent (because the point of 
tangency may be found by (8) ). 
5° . A point, its polar, and three points (14). 
6° . Two points, their polars, and one point on the curve. 
THEOREMS OF GRAVES, MACCULLOGH, AND OTHERS. 
23. (a) Figure 13 represents three biconcyclicconics. Let tangents 
be drawn to the inner one C. By a previous theorem the sects inter¬ 
cepted between the cyclic lines, and the curves will be 
equal, while the point of tangency will bisect the 
portion of the tangent line between the cyclic lines 
and between the curves B and C. In [4] it was shown 
that the two vertical triangles Pab and Pa'b' are 
equal, where aa' and bb' are tangents, and since the 
Fig. 13. point of tangency bisects aa', bb' the triangles must 
approach perfect congruence as aa' approaches bb' , while at the same 
time the vertical triangles cut off by the conic B, will at the same time 
approach congruence. In other words, the line tangent to C cuts off 
a constant space from B, because the increment caused by the tangent 
beginning to move is equal to the decrement, (b) On the contrary, 
considering the conics S and C, a similar reasoning shows that the 
same increment Pmn=P’m'n is added to the space between the conics 
C and A, the fixed tangent M and any variable tangent that is added to 
the space between A, C' (the conic symmetrical to C) and the same 
tangents. In other words the differences between these spaces is equal 
to a constant space. 
1 ° . The reciprocal of the first ( a ) is Graves’ theorem. 
If tangents be drawn, any •point on a conic to a byconfocal conic , the 
sum oj the arc ( convex ) and the tangent lines is constant , i. e., if a 
