684 PROFESSOR TAIT ON 
of that surface. In the statical problem, however, we have 
and thus the corresponding wave-surface has zero for one of its parameters. 
[If this restriction be not imposed, the locus of the point 
ilies xoxB , 
where ¢ is now any given linear and vector function whatever, will be found, 
by a process precisely similar to that just given, to be 
S.(7— $B) ($47) "(7-6 B)=0, 
where ¢ is the conjugate of ¢. This, however, has nothing to do with 
Minp1ne’s Theorem. | 
13. As the reader may not feel secure of results derived by the differentia- 
tion of a vector function operator, it may be well to obtain the result of last. 
section by a more usual process. 
We obviously have by (5”) 
1=78yB +88 
? =o 
SeUr=0. 
Sz 
To make Tz a maximum with these conditions, we have 
or (as in (11 
But also 
SBaB=0 
SBUr=0 
SBB =0 
and, by elimination of 8 and B among these equations, we have as before 
St(atr’) ‘r=0. 
The first of the undifferentiated equations is that of an elliptic cylinder of 
variable magnitude but constant form and position, the second a diametral 
plane, and the third the unit sphere. Obviously there is one maximum and ~ 
one minimum value of Tz. These occur when the variable ellipse given by the 
first and second equation touches the fixed circle given by the second and third. 
