OF ARTS AND SCIENCES. 229 



Pkincipal Diameters. 



For any self-conjugate function q)Q, there are thVee real directions 

 at right angles to each other, and in general only three dii-ections, for 

 which (pQ is parallel to q (Ham. Elem., § 354). We have already 

 seen that (pQ has the direction of the normal at the extremity of q. If, 

 then, Q is in any one of the tliree rectangular directions for which (py 

 is parallel to q, its tangent plane must be parallel to the plane of the 

 other two ; which must therefore bisect all chords parallel to q. These 

 three directions are, therefore, those of a set of conjugate diameters. 



We can see the same thing in a purely analytical way. Let i,j, k 

 repi'esent unit-vectors in the three rectangular directions determined 

 by the above condition ; and let (pi = — c^i, cpj =. — c^j, (pk =z — 

 cjc. Then 



Siqpy = — CgS?)' = 0, 



^jcpk=^ — c^^jk==Q, 



Skqii = — CySki =. 0. 



Conversely, if «, §, y are mutually rectangular, they must be respec- 

 tively parallel to qp«, r^^, q)y. There is thus one set, and in general 

 only one set, of conjugate diameters which are mutually rectangular. 



The- reciprocals of the scalar coefficients q, c^, c^, are the squares of 

 the semiaxes of the quadric ; for 



Sqcpq = T^Sacpa, 



where a is an unit vector in the direction of q. But in the direction 

 of a principal diameter, as i : — 



T-pSccqp« = — T'QSicji = c^^^q = 1. 

 Hence, Tn = -y^, 



and the semiaxis is cfi i. In the same way, the other semiaxes are 

 c~ ij and c^—i k. 



If one of the c's is negative, its square root is imaginary, and there- 

 fore the radius vector does not cut the surface in that direction, and 

 the quadric is a single sheeted hyperholoid. If two ^'s become negative, 

 only one of the principal axes really cuts the surface, which is a dou- 

 ble sheeted hyperboloid. If all three c's are positive, the surface is cut 

 in real points by all three axes, and the quadric is an ellipsoid. But 



