680 PROFESSOR TAIT ON 
become given vectors at right angles to one another, find what the third must 
become that the strain may be pure. The locus of the extremity of the third is, 
for every initial position, one of the single resultants of Minp1ne’s system ; 
and therefore passes through each of the fixed conics. 
Thus we see another very remarkable analogy between strains and couples, 
which is in fact suggested at once by the general expression for the impure 
part of a linear and vector function. 
7. The scalar ¢, which was introduced in equations (7’), is shown by (9) to 
be a function of B alone. In this connection it is interesting to study the sur- 
face of the fourth order 
Star — (ei, + 6)72 —2¢,e,TrSB'r=1 , 
ih 
where T=7 8B. 
But this may be left as an exercise. 
Another form of ¢ (by 7’) is Syy’ + S80’. 
Meanwhile (9) shows that for any assumed value of 6 there are but two 
corresponding Minp1ne lines. 
If, on the other hand, p be given there are in general four values of 8. For 
variety we may take a different mode of attacking equations (7) and (5’), which 
contain the whole matter. In what follows 6 will be merged in p. 
8. Operating by V.8 we transform (5’) into 
p+ BSBp= —(ySy'B+46S88’B) ; (5”) 
Squaring both sides we have 
p?+S*Bp=SBaB (11) 
Since £ is a unit vector, this may be taken as the equation of a cyclic cone; 
and every central axis through the point p lies upon it. For we have not yet 
taken account of (7), which is the condition that there shall be no couple. 
To introduce (7), operate on (5”) by S.y’ and by 8.8’. We thus have, by a 
double employment of (7), 
Sy'p + Sy BSBp =SyaB | (12) 
So’p + Sd’ BSBp =Séah 
Next, multiplying (11) by SBaf, and adding to it the squares of (12), we have 
p SBaB—2SBpSBap—Spap= —SBa’B. ; : (13) 
