OF SIMULTANEOUS ORDINARY DIFFERENTIAL EQUATIONS. 547 
If }, p are quantities such that 
ty = vy + Mito, Yi = Mo + BYo» ete., 
then it is clear that A, = pA. 
Hence, if A = 0, all the determinants of the matrix 
| 1 Ny) C Or 
| Sy, Uy (Gy Oy. | vanish. 
| ane Ma 6 (Gs @ | 
This is the condition that the curve ¢=0 should be cuspidal on the reciprocal 
surface, for the common factor ¢ has to be taken out of the expressions for & 7, @ @, 
and the expressions for the coordinates become 
GIT = Siar fot 4+. . . etc. 
We find, moreover, that 
Lok, + Yons + %lz + Wo; = — WA = O, 
so that the reciprocal surface satisfies the condition 
Ses 3, Ge Ws 
Go tin Gy Or | =O, 
| fy; mM» Ge @,' | 
| wt 
4} 4; 
en er p 
which is of the same form as A = 0. 
Hence this singularity is of the same kind as its reciprocal. The tangents to a 
curve of this kind are true bitangents to the surface, since they meet it in four 
consecutive points, and their reciprocals meet the reciprocal surface in four consecutive 
points. 
Lastly the condition A = 0 is that which must be satisfied if the tangent plane to 
the surface, namely 

Hy, Yo, %, Wy | 
coincides with the osculating plane of the cuspidal curve, that is 
4a 2 
