810 PROFESSOR CHRYSTAL ON THE 



In deducing the above conditions it has been tacitly supposed that p is not infinite. 

 This special case may be treated by change of axes, by special investigation, or by the 

 method of limits. 



We have seen that the tac-locus, if it exists, is furnished by a squared factor in the 

 p-discriminant. The first approximation, 



4:ct 2 (d x 2 + c Q xy + fy) -(bjX+ e x yf = , 



must therefore give a pair of coincident straight lines. The condition for this is 



M/o + e cM - a $ 2 - C W -/<A 2 = ° •* 



This last equation ought, therefore, to be a derivative of b = 0, c =0 when these are 

 satisfied at every point of a branch of the p-discriminant. It will be an interesting test 

 of the accuracy of the foregoing theory to verify that this is actually the case. If we 

 substitute the values of a 2 , &c, calculated above, we find 



Wo + e <M - «2 e o 2 - ^o c i 2 -/<A 2 



=|a 2 (\ 2 + M 2 ) 2 { <f>n>(4>«4>m - tfxa) ~ $**<}>%<, + 2 <p X)l <Ppx<P P y - <P yy <p 2 ia } , 



=|(1 +p 2 Y{<p H (<p xx <j>,„j—<p 2 x ,j) — (p xx (p 2 pu + 2<pxy<t>px4>i>y — Qyytfpx) ' 



Now, from the equations ^ = 0, <p z = 0, <&/ = 0, which are supposed to be satisfied at every 

 point of the branch of the p-discriminant in question, we have 



<p px dx + <f> py dy + <j> pp dp = , 

 <f> xx dx + <j> xy dy + (j> px dp = , 

 $ xy dx + <j> yy dy + <f> py dp = ; 



whence, eliminating dx:dy : dp, we find 



<t>pik<l>xx<pyy — 2 z. v ) ~ (pxxtfpy + %<Pxy<Pp X <t>py ~ <Pyy<j> V = ® > 



which establishes the derivation. 



Geometrical Interpretation of the Conditions for a tac-locus. 



If we regard (x, y, p) as the co-ordinates of a point in space of three dimensions, then 

 the equation (p(x, y, p) = may be taken as representing a surface. The conditions for 

 a tac-point or a tac-locus are therefore simply that the surface <p = have a conical 

 point or a double line. 



Looking at the matter from this point of view, or considering the symmetry of the 

 conditions as regards (x, y, p), we see at once that if the differential equation 



* It may be of interest to note that this is the condition that the function 



a if + (h x + c i>j)P + &<P? + W +fotfi 

 obtained by retaining only such terms of the characteristic of the differential equation as are required to determine an 

 accurate first approximation to the ^-discriminant, is decomposable into factors which are integral and linear in 

 z. V, P- 



