FORM OF TANGENTIAL EQUATION. 
437 
plements of equal angles are equal : therefore the triangles P"DO and CDP are equi- 
angular, and the angle OP"D=PCD ; hence the four points C, P", D, P" are concyclic, 
and therefore the point P'" is the inverse of P" with respect to the imaginary circle the 
square of whose radius is —CO . OD, and whose centre is the point O. 
Again, the angle ODP=FDO=OF"C; therefore the points O, D, P"', P are concyclic, 
and P is the inverse of P'" with respect to the circle whose centre is C, and the square 
of whose radius is the rectangle CD . OD. Hence the proposition is proved. 
Cor. 1. If the point C be at infinity, the point O will coincide with D, and the point 
P will be the reflection of P'" with respect to the line AB. 
Cor. 2. If the points A, B, C, O be the centres of inversion of a bicircular quartic, 
and if the point P be on the curve, the points P', P", P'" will also be on the curve. 
141. Let the radii of the generating circles of the bicircular quartic which touch it 
at the four pairs of points (PF), (P'P"), (F'F"), (P"'P) be denoted by g, respec- 
tively. Let the angle which the line APP' makes with any fixed line in the plane, say 
the axis of x, be denoted by Q, and the angles which the lines BP'P", OP"P'", CPP'" 
make with the same line by Q', S", S'". 
Now if the points P, P', P", F" describe infinitesimal arcs, we have (see art. 139), 
denoting these arcs by ds, ds', See., 
ds' —ds =2% dQ , 
ds' -ds" =2§' d& , 
ds" +ds"'=2 ? "d&' , 
ds"'—ds =2 fdff". 
Hence 
ds'= s dQ+ s 'd0'+g"dQ"+g w dQ"'; (294) 
s >=fedQ+fe'dQ'+§ s "dQ"+fe'"d"' (295) 
Hence the arc of a bicircular quartic is the sum of four similar integrals. We shall 
find that each of them is expressed in terms of elliptic integrals. This theorem is our 
generalization of Roberts’s and Genocchi’s theorems*. 
* The following proof of the theorem art. 138 will apply equally to sphero-quartics, and will lead to an 
important extension of the theorem of this article : — 
Let CY, C'V' he two consecutive tangents to the focal conic F of the bicircular quartic (or, in the case of a 
sphero-quartic, to the focal sphero-conics), and OPP', OQQ' two perpendiculars to CY, CY' (see fig. art. 130). 
If CY, C'V' intersect the generating circle in the points R, R', it is evident, from geometrical considerations, that 
RR'=!(P'Q'-PQ). 
But RR'=fcZ0 for bicircular quartics and =sin ^dO for sphero-quartics ; hence, remembering the theorem in the 
footnote to art. 140, we have, for sphero-quartics, 
s'=J sin sin g'dO' +J sin fdd" + j* sin f'dO'", 
and the rectification of sphero-quartics is reduced to elliptic integrals . — November 1877. 
