227] ON THE THEORY OF RECIPROCAL SURFACES. 27 
to which I annex also, without transformation, the four equations for q, r, q', r', viz.: 
q =b 2 — b — 2k — — (it, 
r = c 2 — c —2 h — 3/3, 
q' = b' 2 -b'~ 2k' - 3 7 ' - W, 
r' = 6' 2 - c' - 2h' - 3/3, 
the last two of which, neglecting singularities, give 
= 4n — 2) (n — 3) (n 2 + 2n — 4), 
r' = 2w (n — 2) (3?i — 4), 
which are the values given by Mr Salmon. I remark, in conclusion, that there is 
considerable difficulty in the geometrical conception of the points i and the planes %, 
and the subject appears to require further examination. In the case of a surface of 
the order n without multiple lines, we have not only i = 0 (which is a matter of 
course), but also % = 0. In my paper before referred to, I showed, or attempted to 
show, by geometrical reasoning, that the common tangent planes of the spinode 
develope and the node-couple develope are stationary planes of the one or the other 
of the two developes, that is, ¿' = 0, and the reasoning seems correct as far as it goes, 
but it was not shown how the demonstration would (as it ought to do) fail in the 
case of a surface having a double or cuspidal curve. I showed also that in the case 
where the common tangent plane is a stationary plane of the spinode develope (that 
is for the planes /3'), the spinode curve and the node-couple curve touch instead of 
simply intersecting; it would seem that the tangent plane at such point is to be 
counted once, and not twice, in reckoning the number /3' of such tangent planes: the 
like remark applies, of course, also to the points of intersection B of the double and 
cuspidal curves. 
4—2