376 
MR. W. SPOTTISWOODE ON THE FORTY-EIGHT 
and in the U' coordinates ; and as we are concerned with the ratios only of the 
coefficients and not with their absolute values, we are in fact concerned only with the 
ratios of the U coordinates inter se, and of the U / coordinates inter se, and not with 
their absolute values. Hence the number of independent coordinates will be reduced 
to 48 — 34 — 2=12, as it should be. 
§ 1. Formation of the equations. 
With a view to the problem in question, it is first required to form the equations 
of two quadrics having a generating line in common. For the present purpose the 
following appears the simplest way of effecting this ; let 
u=ax-}-/3y-\-yz-\-$t, u = ate + fi'y-\-y'z-\-S't, .(1) 
v =oL 1 x+/3 1 y+y ] z+‘8 l t, v'=afx +/3 1 h/+ 7i z + S i'* > 
w = a.Tpc +/3q/-f- y°z +Sot, w = afx -f- fi.iy + y(z +S 2 't, 
From these we may form the three equations 
viv'—vw=(a, b, . . )(x, y, z, t) 2 = 0 ,.( 2 ) 
ivu'—w'u=(a 1 , b lt . . )(x, y, z, tf— 0, 
uv — u'v — (%, 6 3 , . . )(x, y, z, t) 3 = 0, 
of which two only, of course, are independent. Any two of them may be taken as 
representing the two quadrics in question. Thus, if we take the last two, the equa¬ 
tions of the common generating line will be u— 0, u'— 0. 
The next step is to eliminate the four variables in turn from the two quadrics ; for 
which purpose it will be convenient to express the system (2) in the following form : 
u \u =v : v —w : w ;.(3) 
or, introducing an indeterminate quantity 9, we may use, instead of equations (3), the 
following, viz. : 
u-\-9u'= 0, v -f- 6v' = 0 , w-\-9w' = 0 . (4) 
In order to eliminate one of the variables, say t, let us write, 
u=Ut-\-Bt } a'—u{ -f-S f, . (5) 
v=?;/+S F, F=v/+Si 't, 
w=w,-\- S.Y, w'='W/ ,J r S.A. 
