535 
of Edinburgh, Session 1881-82. 
of Professor Tait’s paper), and g depending on h through (22) which 
becomes 
(25) g 2 + '2h = M 2 - p 2 . 
The equation of the surface will he 
( S p&rp — M 2 p 2 + 
1 + 2^S p4>i'<f>j' ) 
and the elimination of g by (25) will raise the degree in p to the 4th. 
This equation is concordant with that numbered (10) of Professor 
Tait’s paper : this last one is transformable into ours (26) by the 
elimination of t (which is our - m 0 ). The elimination can be 
founded on the equation (9) of the quoted paper, and by treating 
by S . the equation (8') so as to give (in our notation) 
hm 1 - m 2 Sp<f>i , <l>f + = 0 . 
The value resulting for t (or of our m 2 ) will be in our notation 
m 2 n = H , 
n containing then a double signed radical, but, as we shall find, the 
sign of it will be definite, and (in anticipation to our own use) we 
state that we shall have 
(27) n — gh^r S p<f>i'<f>j' 
H = h 2 - M 2 h + Mj . 
If, on the contrary, we wish to form the equation of the locus of 
the feet of perpendiculars from the origin on the single resultants, 
then we are no more free to establish a relation between the para- 
meters of p( )q and u. The condition of the perpendicularity 
between the line (7) and p itself will be 
S&> = 0. 
The equation (7) will give its first scalar relation 
(28) u = 0. 
Then (21) with this value of u will constitute a second scalar 
relation (equivalent with 7t /2 = - 1). 
A third scalar relation will be obtained in treating by S( )<f>i'<bf 
not (7) itself, but a transformation of it derived from (15). Finally, 
