546 PROFESSOR A. C. DIXON ON THE SINGULAR SOLUTIONS 
where 2%, Yo.-- %, Y,;--. are functions of a second variable uw, such that all the deter- 
minants of the matrix 
| 2, Yr 2 Wr 
| ©, Yor 2 Wo 
a R999 opp 0 
| Ly, Yor > Wo 
vanish, dashes being used to indicate differentiation with respect to uw, and dots 
differentiation with respect to ¢. 
If € », ¢, w are the determinants of the matrix 
a, Oh, By UW |? 
/ / f? / 
| “x,y, z, W 
then the reciprocal surface is the locus of (€, », ¢, @). 
We will now find whether the curve ¢ = 0 on the reciprocal surface can be cuspidal. 
If € »,  » are expanded in powers of t we have at once & = 0, yn, = 0, & = 0, 
w) = 0, €, denoting the coefficient of ¢” in €, and so on, 
We also find 
ay & + Yo T % 6 + wo, = 0, 
Ly Eo + Yo Nz H 2% & + Wy = 0,7 
XE, + Yom +O + wo, = 0, 
ty fo + Yo Ns + % & + Wy @, = 2 
mY / post / — 
Lo» Yor % >» Wy | = 2A, say. 
X, Yo, %, Wo 
| 
©, Yo, 2, Wa | 
| 
aw, On Bee 
| My, Yrs > Wy 
and hence, differentiating the first and using the third, 
Hoey + Yom’ + 2oby + wow,’ = 0. 
Also 
‘t ” myth? ; Ww = ee ed? LL v/ piss 
®y'E Yo M + 2% G + Wy oy = 2 | “0 » Yo »% » >, | = — 2A, say, 
| Zo Yor 20» Wos | 
| | 
®, Yo, %, We, | 
igen tory , , | 
| 4 > Yo> % > Wo >» 
so that 
Go sh Yor =F BiG + Wy Wy, — ON 
